Second-Order Nonlinear Equations

B.5. Second-Order Nonlinear Equations

B.5.1. Parabolic Equations

2 B.5.1-1. Equations of the form ✁

Exact solution with multiplicative form: à

is an arbitrary constant and the function Õ ( ) is determined by solving the ordinary differential equation

1 å . Exact solution:

, ) = exp Ó✟✞ ( ) Ö + ✠ ( ) Ø ,

where the functions ✁

( ) and ✠ ( ) are given by

, and ✡ and ☞ are arbitrary constants. å

2 ✝ . Exact solution:

, ) = exp ✒✔✓ ( ) ✑ 2 + ö ( ) ✕ ,

where ✁

( ) and ö ( ) are given by

, and ✡ and ☞ are arbitrary constants.

3 ✝ . There are also exact solutions of the more general form å

, ) = exp ✒✔✓ 2 ( ) ✑ 2 + ✓ 1 ( ) ✑ + ✓ 0 ( ) ✕ ,

2 ( ), ✓ 1 ( ), and ✓ 0 ( ) are determined by a system of ordinary differential equations that can be integrated. ê⑨ë

where the functions ✁

Reference : V. F. Zaitsev, A. D. Polyanin (1996). 2 ✂

2 + ✛ ( ) ln +[ ✜ ( ) + ì ( ) + s( )] . ✙ ✙ ✚ ✚ ✙

Exact solution: ✚

, ) = exp ✒ ✓ ( ✑ 2 2 ) + ✓ 1 ( ) ✑ + ✓ 0 ( ) ✕ ,

where the functions ✁

( )( ✤ = 1, 2, 3) are determined by solving the following system of first-order ordinary differential equations with variable coefficients: ✏

(the arguments of , ✗ , í ✁ , and ✦ are not specified and the prime denotes the derivative with respect to ).

The change of variable ✚ = exp ★ leads to an equation of the form B.5.1.14,

which has exponential and sinusoidal solutions in ✑ .

B.5.1-2. Equations of the form ✢

Exact solution: ✚

, ) = exp ✒✔✓ ( ) ✑ 2 + ö ( ) ✕ ,

where the functions ✁

( ) and ö ( ) are determined by solving the following system of first-order ✏

ordinary differential equations with variable coefficients (the arguments of and ✗ are not specified):

Integrating successively yields

, where ✡ and ☞ are arbitrary constants.

Exact solution: ✚

, ) = exp ✒ ✑ ✓ ( )+ ö ( ) ✕ ,

where the functions ✁

( ) and ö ( ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients:

(2) Integrating first (1) and then (2), we obtain ( è 1 and è 2 are arbitrary constants)

Exact solution: ✚

( ✑ , ✶ ) = exp ✒✔✑ 2 ✓ ( ✶ )+ ✑ ✽ ( ✶ )+ ✾ ( ✶ ) ✕ , ( ✑ , ✶ ) = exp ✒✔✑ 2 ✓ ( ✶ )+ ✑ ✽ ( ✶ )+ ✾ ( ✶ ) ✕ ,

Exact solution: ✚

, 2 ✶ ) = exp ✒✔✓ ( ✶ ) ✑ + ✽ ( ✶ ) ✕ ,

where the functions ✓ ( ✶ ) and ✽ ( ✶ ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients: ✏ ✵

Exact solution: ✚

where the functions ✓ ( ✶ ), ✽ ( ✶ ), and ✾ ( ✶ ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients:

Exact solution in additive form: ✚

Here, ❆ is an arbitrary constant and the function ❋ ( ● ) is determined by the nonlinear ordinary differential equation

By changing variable ❍

= ❃ ✽ this equation is reduced to the second-order linear equation

Exact solution in additive form: P

The equation has exact solutions of the form P

( ● , ❨ )= ❋ ( ❨ )+ ❝ ( ❨ ) exp( ❞ ● ), where ❞ is a root of the quadratic equation ❴▲❞ 2 + ➳ ❑❡❞ + = 0.

Exact solution: ❘

where the functions ❋ ( ❨ ), ❝ ( ❨ ), and ❢ ( ❨ ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients:

(3) Equation (1) for ❋ is a Bernoulli equation, which is easy to integrate. After that, equations (2)

and (3), which are linear in ❝ and ❢ , are integrated successively. As a result, we find

where ❤ 1 , ❤ 2 , and ❤ 3 are arbitrary constants. A degenerate solution with ❋ ≡ 0 corresponds to the ♥♣♦ limit case ❤ 1 ❧ ♠

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

1 ❘ . Exact solution:

(1) where the functions ❋ ( ❨ ) and ❝ ( ❨ ) are determined by solving the following first-order ordinary ❪

( ● , ❨ )= ❋ ( ❨ )+ ❝ ( ❨ ) exp rts ● ✉ − ❴ ✈ ,

differential equations with variable coefficients (the arguments of ▼ ❪ , ❵ , and are not specified):

(3) Equation (2) for ❋ ( ❨ ) is a Riccati equation; it can be reduced to a second-order linear equation. ❪

Many solutions of equation (2) for various ▼ , ❵ , and can be found in Kamke (1977) and Polyanin and Zaitsev (1995).

Whenever a solution of equation (2) is known, the solution of equation (3) for ❝ ( ❨ ) can be evaluated from

( ❨ )= ❩ exp ❥ − ❀ ❴❄❨ + (2 ❴▲▼ ❋ + ❵ ) ❊ ❨✸❦ ,

where ❩ is an arbitrary constant.

. Exact solution of a more general form: ( ● , ❨ )= ❋ ( ❨ )+ ❝ ( ❨ ) ✇✔❤ exp r✬● ✉ − ❴ ✈ + ① exp r − ● ✉ − ❴ ✈▲② ,

< 0, where the functions ❋ ( ❨ ) and ❝ ( ❨ ) are determined by solving the following system of first-order

ordinary differential equations with variable coefficients:

= ❴▲▼ r✬❋ 2 +4 ❤ ① ❝ 2 )+ ❵ ❋ + ,

(5) One can express ❋ in terms of ❝ from (5) and then substitute into (4). As a result, one obtains a ❪

second-order nonlinear equation for ❝ ; if ▼ , ❵ , = const, this equation is autonomous and, hence, admits reduction of order. ❳

3 q . Exact solution ( ❑ is an arbitrary constant):

(6) where the functions ❋ ( ❨ ) and ❝ ( ❨ ) are determined by solving the following system of first-order

( ● , ❨ )= ❋ ( ❨ )+ ❝ ( ❨ ) cos r✬● ✉ ❴ + ❑❡✈ ,

ordinary differential equations with variable coefficients:

= ❴▲▼ r✬❋ 2 + ❝ 2 )+ ❪ ❵ ❋ + ,

(8) One can express ❋ in terms of ❝ from (8) and then substitute into (7). As a result, one obtains a ❪

second-order nonlinear equation for ❝ ; if ▼ , ❵ , = const, this equation is autonomous and, hence, admits reduction of order. ♥♣♦

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

The equation has exact solutions of the form P

( ● , ❨ )= ❋ ( ❨ )+ ❝ ( ❨ ) exp( ❞ ● ), where ❞ is a root of the quadratic equation ❞ 2 + ❴▲❞ + ❑ = 0.

Exact solution in additive form: P

where ❩ is an arbitrary constant and the function ⑨ ( ⑩ ) is determined by the following second-order ordinary differential equation with variable coefficients:

Exact solution: ❻

where the functions ⑨ ( ❨ ), ❝ ( ❨ ), and ❢ ( ❨ ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients: ❪ ⑥

(3) Equation (1) for ⑨ ( ❨ ) is a Riccati equation; it can be reduced to a second-order linear equation.

For solutions of Riccati equations, see Kamke (1977) and Polyanin and Zaitsev (1995). Whenever a solution of equation (1) is known, the solutions of equations (2) and (3) can be obtained successively (the equations are linear in ❝ and ❢ ).

Exact solution in additive form: ❻ ❳ ❹ ❻ ❘

where ❩ is an arbitrary constant and the function ⑨ ( ⑩ ) is described by the second-order ordinary differential equation

B.5.1-3. Equations of the form 2

Exact solution: ❘

( ⑩ , ❨ ) = exp ✇ ⑨ ( ❨ ) ⑩ 2 + ❝ ( ❨ ) ② ,

where the functions ⑨ ( ❨ ) and ❝ ( ❨ ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients (the arguments of ❸ and ❵ are not specified):

Integrating successively yields

and ① are arbitrary constants.

Exact solution: ❻

( ⑩ , ❨ ) = exp ✇ ⑨ ( ❨ ) ⑩ 2 + ❝ ( ❨ ) ② ,

where the functions ⑨ ( ❨ ) and ❝ ( ❨ ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients:

(2) The Riccati equation (1) for the function ⑨ ( ❨ ) can be reduced to a second-order linear equation.

For solutions of the Riccati equation, see Kamke (1977) and Polyanin and Zaitsev (1995). On solving (1), one can determine the solution of the linear equation (2) for ❝ ( ❨ ).

ln ❺ + ❱ ❲ ❺

Exact solution: ❘

( ❨ ) ❬ − ) = exp ➉ + ❝ ( ❨ ) ② ,

where the functions ⑨ ( ❨ ) and ❝ ( ❨ ) are determined by the ordinary differential equations

Integrating yields

where ❤ and ① are arbitrary constants.

Exact solution: ❳

, ❨ ) = exp ✇ ❤ ❬ ➋ + ⑨ ( ⑩ ) ② ,

where ❤ is an arbitrary constant and the function ⑨ ( ⑩ ) is determined by the ordinary differential equation

Exact solution in multiplicative form: ❻

, ❨ ) = exp ❥t❩ ❬ ➋ + ❬ ➋

where ❩ is an arbitrary constant and the function ⑨ ( ⑩ ) is determined by the ordinary differential equation

Exact solution in multiplicative form: ❻

, ❨ ) = exp ❥✬❩ ❬ ➋ + ❬ ➋

where ❩ is an arbitrary constant and the function ⑨ ( ⑩ ) is determined by the ordinary differential equation

Exact solution in additive form: ➍

where ➑ is an arbitrary constant and the function ⑨ ( ⑩ ) is determined by the following second-order ordinary differential equation:

B.5.1-4. Equations of the form 2

2 + ❸ ➣t⑩ ➐

Exact solution: ➏

where ➝ , ➞ , and ➡ 0 are arbitrary constants.

1 ➭ . Exact solution: ➏

where ➝ and ➞ are arbitrary constants.

2 ➭ . Exact solution:

where ➝ , ➞ , and ➑ are arbitrary constants.

Exact solution: ➏

where the functions ➵ ( ➐ ), ➼ ( ➐ ), and ➻ ( ➡ ) are described by ordinary differential equations

where ➑ is an arbitrary constant. Integrating successively yields ➠ ➠ ➠

where ➝ 1 , ➝ 2 , ➞ 1 , and ➞ 2 are arbitrary constants.

+ ➫ ( ) ➦ ➧ + ➺ ( ) + s( ).

Exact solution: ➩

where the functions ➵ ( ➐ ), ➼ ( ➐ ), ➘ ( ➐ ) are determined by solving the following system of first-order ➓

ordinary differential equations with variable coefficients (the arguments of ➤ , → , , and ➴ are not specified):

(3) Equation (1) for ➵ = ➵ ( ➐ ) is a Bernoulli equation; it is easy to integrate. After that, one can

successively construct the solutions of equations (2) and (3); each equation is linear in the unknown function. ➷♣➬

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

. Exact solution:

where the functions ➍

( ➐ ) and ➼ ( ➐ ) are determined by solving the following first-order ordinary differential equations with variable coefficients (the arguments of ➤ and → are not specified):

(3) Equation (2) for ➵ = ➵ ( ➐ ) is a Riccati equation; it can be reduced to a second-order linear

equation. The books by Kamke (1977) and Zaitsev and Polyanin (1995) present many solutions of equation (2) for various ➤ and → .

Given a solution of equation (2), the solution of equation (3) for ➼ = ➼ ( ➐ ) is evaluated by

(4) where ➑ is an arbitrary constant.

( ➐ )= ➑ exp ➲ ( ➸ ✃ 2 ➟ ➵ +2 ➜▲➵ + ➤ ) ➥ ➐✸➳ ,

2 ➭ . Exact solution ( ➝ ➏ is an arbitrary constant):

, (5) where the functions ➍

( ➡ , ➐ )= ➵ ( ➐ )+ ➼ ➐ ) cosh( ➡ + ➝ ), ✃

( ➐ ) and ➼ ( ➐ ) are determined by solving the following first-order ordinary differential equations with variable coefficients (the arguments of ➤ and → are not specified):

(7) One can express ➵ from (7) in terms of ➼ and substitute the resulting ➵ into (6). As a result, one

arrives at a second-order nonlinear equation for ➼ (if ➤ , → = const, this equation is autonomous and, hence, admits reduction of order).

3 ➭ . Exact solution ( ➝ ➏ is an arbitrary constant):

, where the functions ➍

( ➐ ) and ➼ ( ➐ ) are determined by solving the system of first-order ordinary differential equations

4 ➭ . Exact solution ( ➝ is an arbitrary constant): ➏

( ➡ , ➐ )= ➵ ( ➐ )+ ➼ ( ➐ ) cos( ✃ ➡ + ➝ ), ✃ = ➹ ➸

where the functions ➍

( ➐ ) and ➼ ( ➐ ) are determined by solving the system of first-order ordinary differential equations

(10) One can express ➵ from (10) in terms of ➼ and substitute the resulting ➵ into (9). Thus, one

arrives at a second-order nonlinear equation for ➼ (if ➤ , → = const, this equation is autonomous and, hence, admits reduction of order). ➷♣➬

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

The equation has an exact solution of the form ➩

where the functions ➵ ( ➐ ), ➼ ( ➐ ), and ➘ ( ➐ ) are determined by a system of first-order ordinary differential equations with variable coefficients (the system is not specified here). ➏ ➏

B.5.1-5. Equations of the form ❮ ❰ ❾ = ➸ ❮ ➾ ➤ ( ) ❮ ❰ ➚ + ➤ ➣✬➡ , ➐ , , ❮ ❰ ↔ .

The change of variable ➍

leads to an equation of the form B.5.1.29,

+ ➤ ( Õ ), which admits solutions with the form Ð = ➵ ( Õ ) ➡ 2 + ➼ ( Õ ) ➡ + ➘ ( Õ ).

The change of variable ➩

leads to an equation of the form B.5.1.29,

➷♣➬ which admits solutions with the form

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

For ➩

= 0, see equation B.5.1.33. The change of variable Ð = Ø

leads to an equation of the form B.5.1.30,

( Õ ), which admits solutions with the forms

where the functions ➵ ( Õ ) and ➼ ( Õ ) are determined by a system of first-order ordinary differential equations; the parameter ✃ is a root of a quadratic equation and Û ➷♣➬ is an arbitrary constant.

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

The change of variable ➩

= ß à ❰ leads to an equation of the form B.5.1.27,

which admits solutions with the form Ð = ➵ ( Õ ) ➡ 2 + ➼ ( Õ ) ➡ + ➘ ( Õ ).

The change of variable ➩

= ß à ❰ leads to an equation of the form B.5.1.26,

which admits solutions with the form Ð = ✃ ( Ú❄➡ + ➜ ) Õ + ➵ ( ➡ ).

= 0, see equation B.5.1.35. The change of variable Ð = ß à ❰ leads to an equation of the form B.5.1.30,

which admits solutions with the forms

where the functions ➵ ( Õ ) and ➼ ( Õ ) are determined by a system of first-order ordinary differential equations; the parameter â ➷♣➬ is a root of a quadratic equation and Û is an arbitrary constant.

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

B.5.1-6. Equations of the form 2

Exact solution: ➩

where the functions ➵ ( ➡ ) and ➼ ( ➡ ) are determined by the ordinary differential equations

The first equation can be treated independently. The second equation has a particular solution ➠ ➠ ➠ ➠

( ➡ )= ➵ ( ➡ ), and hence, its general solution is given by

where Û 1 and Û 2 are arbitrary constants.

Exact solution: ➩

where the functions ➵ = ➵ ( ➡ ) and ➼ = ➼ ( ➡ ) are determined by the first-order ordinary differential equations

Integrating yields Ñ

, where æ ➷♣➬ and ç are arbitrary constants.

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

Exact solution in additive form: ➩

( ë , )=− ln(

+ Û )+ é ln ➲➊ê

where æ , ç , and Û are arbitrary constants.

B.5.1-7. Equations with three independent variables.

+ ï ln ï .

Exact solution in multiplicative form: î

is an arbitrary constant and the function ➻ ( , ö ) satisfies the stationary equation

+ ù ➻ ln ➻ = 0.

Incomplete separable exact solution (the solution is separable in the space coordinates ☞ and ✌ but not in time ✍ ):

The functions ✏ ( ☞ , ✍ ) and ✑ ( ✌ , ✍ ) are determined from the one-dimensional nonlinear parabolic differential equations

where ✖ ( ✍ ) is an arbitrary function.

B.5.2. Hyperbolic Equations

B.5.2-1. Equations of the form ✙

Exact solution in multiplicative form:

where the functions ✏ ( ✍ ) and ✑ ( ☞ ) are determined by the second-order ordinary differential equations ✗

where ✖ is an arbitrary constant.

Exact solution in additive form:

where ✴ is an arbitrary constant and the functions ✏ = ✏ ( ✍ ) and ✲ = ✲ ( ✳ ) are determined by solving the second-order ordinary differential equations

(2) The general solution of equation (1) is given by

( )= 1 cosh( )+ 2 sinh( )+

1 cos( )+ 2 sin( )+

( ✻ ) sin[ ✹ ( ✍ − ✻ )] ✼ ✻

if ✵ =− ✹ < 0,

where ✖ 1 and ✖ 2 are arbitrary constants.

( ✲ )= ✲ ✧ ✷ ✤ , which leads to a first-order linear equation.

Equation (2) can be solved with the change of variable 2

Exact solution in additive form:

1 cosh( ✹ ✍ )+ ✖ 2 sinh( ✹ ✍ ) if ✵ = ✹ > 0,

( ✍ )= ✖ 1 cos( 2 ✹ ✍ )+ ✖ 2 sin( ✹ ✍ ) if ✵ =− ✹ < 0, where ✖ 1 and ✖ 2 are arbitrary constants, and the function ✏ ( ☞ ) is determined by the ordinary

differential equation

Exact solution:

( ☞ , ✍ )= ✏ ( ✍ )+ ✑ ( ✍ ) exp( ✴ ☞ ),

where ✴ is a root of the quadratic equation ✫✶✴ 2 + ✵✛✴ + ✹ = 0 and the functions ✏ ( ✍ ) and ✑ ( ✍ ) are determined by the following system of second-order ordinary differential equations: ✓ ✗

(2) In the special case ( ✍ ) = const and ( ✍ ) = const, equation (1) is autonomous and has particular

solutions of the form ✏ = const and, hence, can be integrated in quadrature. Equation (2) is linear in ✑ , and consequently, with ✏ = const, its general solution is expressed in terms of exponentials or sine and cosine.

Exact solution:

(1) where the functions ✏ ( ✍ ), ✑ ( ✍ ), and ✿ ( ✍ ) are determined by solving the following system of second- ✓ ✗

order ordinary differential equations with variable coefficients (the arguments of , , and ✕ are not specified):

(4) Equation (2) has the trivial particular solution ✏ ( ✍ ) ≡ 0; the corresponding solution (1) is linear

in the coordinate ☞ . Equation (3) has a particular solution ✑ =¯ ✏ ( ✍ ), where ¯ ✏ ( ✍ ) is any nontrivial particular solution of equation (2). Hence, the general solution of equation (3) is given by

where 1 and are arbitrary constants. If the functions and proportional, then =−

( ✏ = const) is a particular solution of equation (2). Equation (4) linear in ✿ = ✿ ( ✍ ).

Exact solution in additive form:

Here, ❁ , ❂ , and ✖ are arbitrary constants, and the function ✏ ( ☞ ) is determined by solving the second-order nonlinear ordinary differential equation ✓ ✣ ✗

Exact solution in additive form:

Here, the functions ✏ ( ✍ ) and ✑ ( ☞ ) are determined by solving the second-order ordinary differential equations

The general solution of the first equation is given by

( ✍ )= ✖ 1 cosh( ✹ ✍ )+ ✖ 2 sinh( ✹ ✍

( ✍ )= ✖ 1 cos( ✹ ✍ )+ ✖ 2 sin( ✹ ✍

where ✖ 1 and ✖ 2 are arbitrary constants.

1 ❄ . Exact solution:

where the functions ✤

( ✍ ) and ✑ ( ✍ ) are determined by solving the following second-order ordinary ✓ ✗

differential equations with variable coefficients (the arguments of ✓ ✗ , , and ✕ are not specified):

2 ❄ . Exact solution of a more general form:

and ❂ are arbitrary constants and the functions ✏ ( ✍ ) and ✑ ( ✍ ) are determined by the following system of second-order ordinary differential equations with variable coefficients (the arguments of ✓ ✗

, , and ✕ are not specified):

3 ❄ . Exact solution:

( ☞ , ✍ )= ✏ ( ✍ )+ ✑ ( ✍ ) cos ☞ ❆ ✫ + ✖ ✤ ,

Exact solution in additive form:

Here, ❁ , ❂ , and ✖ are arbitrary constants and the function ✏ ( ☞ ) is determined by solving the second-order nonlinear ordinary differential equation

Exact solution in additive form:

Here, the functions ✏ ( ✍ ) and ✑ ( ☞ ) are determined by the second-order ordinary differential equations

The general solution of the first equation is given by

( )= 1 cosh( )+ 2 sinh( )+

) sinh[ ✹ ( ✍ − ✻ )] ✼ ✻

if ✫ = ✹ > 0,

( ✍ )= ✖ 1 cos( ✹ ✍ )+ ✖ 2 sin( ✹ ✍

where ✖ 1 and ✖ 2 are arbitrary constants.

Exact solution in multiplicative form:

where ✴ is an arbitrary constant and the function ✏ ( ✍ ) is determined by the second-order linear ordinary differential equation

B.5.2-2. Equations of the form ✙

Exact solution for ● ≠ 2:

= ✪ 1 4 ✜ (2 − ● 2 2 2− ) ■ ( ✍ + ✖ ) −( ☞ + ❍ ) ✭ 2(2− ■ ) ✎ ✎ , where ✖ is an arbitrary constant and the function

= ( ✽ ) is determined by the generalized Emden–Fowler equation

A number of exact solutions to equation (1) for some specific = ( ) can be found in Polyanin and Zaitsev (1995). In the special case ● = 1, the general solution of equation (1) is given by

( ) , where ✖ 1 and ✖ 2 are arbitrary constants.

Exact solution for ● ≠ 2: ✎

where ✖ is an arbitrary constant and the function = ( ✳ ) is determined by the ordinary differential equation

For ❁ ≠ 1, the change of variable ✳ = ✹ ✽ 1− ▼ ( ✹ =

1) brings this equation to the generalized Emden–Fowler equation

whose solvable cases are presented in Polyanin and Zaitsev (1995).

Exact solution for ● ✎ ≠ 2: ✎

= 1, where ✖ is an arbitrary constant and the function = ( ✽ ) is determined by the ordinary differential

The change of variable ◆ ( )= ✽ ✧ ❏ leads to a first-order separable equation. Integrating this equation yields the general solution in implicit form: ✎

where ✖ 1 and ✖ 2 are arbitrary constants.

Exact solution for ✎ ● ≠ 2: ✎

= 1, where ✖ is an arbitrary constant and the function = ( ✽ ) is determined by the ordinary differential

Exact solution for ✴ ✎ ≠ 0: ✎

= 1, where ✖ is an arbitrary constant and the function = ( ✽ ) is determined by the ordinary differential

(1) For ✫ = ✜ ✴ , the solution of equation (1) is given by

( ) ✼ , where ✖ 1 and ✖ 2 are arbitrary constants.

2 ✜ ✴ , the change of variable ✳ = ✽ ❙ ❈ brings (1) to the generalized Emden–Fowler equation

whose solvable cases are presented in Polyanin and Zaitsev (1995).

Exact solution for ✴ ✎ ≠ 0: ✎

= 1. Here, ❁ is an arbitrary constant and the function = ( ✽ ) is determined by the ordinary differential

which by the change of variable ✎

( )= ✽ ❏ ✧ is reduced to a separable first-order equation. Integrating this equation yields the general solution in implicit form: ✎

1 and ✖ 2 are arbitrary constants.

+ ❖ P 2 ◗ 2 ( ) ✄ + ( ).

Exact solution for ✴ ✎ ≠ 0: ✎

= 1, where ✖ is an arbitrary constant and the function = ( ✽ ) is determined by the ordinary differential

equation

B.5.2-3. Other equations.

1 ❄ . Exact solutions: ✎

1 , ✖ 2 , ✖ 3 , ✖ 4 , and ✜ are arbitrary constants.

2 ❄ . Exact solution:

where the functions ✏ = ✏ ( ✍ ), ✑ = ✑ ( ✍ ), and ✿ = ✿ ( ✍ ) are determined by the system of ordinary differential equations

3 ❄ . Exact solution in multiplicative form: ✎

where the functions ❚ = ❚ ( ✍ ) and ❯ = ❯ ( ☞ ) are determined by the ordinary differential equations ( ✖ is an arbitrary constant)

The latter equation is autonomous and has a particular solution ❯ = 1 6 ✖ ☞ 2 and, hence, is integrable in quadrature.

Exact solution:

where the functions ✏ = ✏ ( ✍ ), ✑ = ✑ ( ✍ ), and ✿ = ✿ ( ✍ ) are determined by the system of ordinary differential equations

B.5.3. Elliptic Equations

B.5.3-1. Equations of the form ✙

Exact solution in multiplicative form: ✎

where the functions ✏ ( ☞ ) and ✑ ( ✌ ) are determined by the ordinary differential equations ✓

where ✖ is an arbitrary constant.

2 + ✟ 2 = ( ) ln +[ ✥ ( ) + ( )] .

Exact solution in multiplicative form: ✎

where the function ✏ ( ☞ ) is determined by the ordinary differential equation ✓ ✗

Exact solution:

(1) Here, the functions ✏ ( ☞ ), ✑ ( ☞ ), and ✿ ( ☞ ) are determined by the following second-order ordinary ✓ ✗

differential equations with variable coefficients (the arguments of ✓ ✗ , , and ✕ are not specified):

(4) If a solution ✏ = ✏ ( ☞ ) of the nonlinear equation (2) is found, then the functions ✑ = ✑ ( ☞ ) and

= ✿ ( ☞ ) can be determined successively from equations (3) and (4), which are linear in ✑ and ✿ . By comparing equations (2) and (3), one can see that equation (3) has a particular solution

= ✏ ( ☞ ). Hence, the general solution of (3) is given by (see Polyanin and Zaitsev 1995)

Note that equation (2) has the trivial particular solution ✓ ✏ ( ☞ ) ≡ 0, to which there is a corresponding ✗ ✗ ❀ ✓

solution (1) linear in the coordinate ✌

. If the functions 1 and are proportional, then ✏ =−

( ✏ = const) is a particular solution of equation (2).

1 ❄ . Exact solution:

where the functions ✤

( ☞ ) and ✑ ( ☞ ) are determined by the following second-order ordinary differential ✓ ✗

equations with variable coefficients (the arguments of ✓ ✗ , , and ✕ are not specified):

2 ❄ . Exact solution of a more general form: ✎ ✣

< 0, where the functions ✏ ( ☞ ) and ✑ ( ☞ ) are determined by the following system of second-order ordinary

( ☞ , ✌ )= ✏ ( ☞ )+ ✑ ( ☞ ) ✪ ❁ exp ✌ ❆ − ✫ ✤ + ❂ exp − ✌ ❆ − ✫ ✭ ✤ ,

differential equations with variable coefficients: ✓ ✣

3 ❄ . Exact solution:

> 0, where ✖ is an arbitrary constant and the functions ✏ ( ☞ ) and ✑ ( ☞ ) are determined by the following

( ☞ , ✌ )= ✏ ( ☞ )+ ✑ ( ☞ ) cos ✌ ❆ ✫ + ✖ ✤ ,

system of second-order ordinary differential equations with variable coefficients: ✓ ✣ ✗

Reference : V. F. Zaitsev, A. D. Polyanin (1996).

Exact solution in additive form:

Here, the functions ✏ ( ☞ ) and ✑ ( ☞ ) are determined by solving the second-order ordinary differential equations

is an arbitrary constant.

B.5.3-2. Equations of the form ✎

1 ❄ . For ● ≠ 2 and ❬ ≠ 2, there are exact solutions of the form ✎ ✎

. Here, the function = ( ✳ ) is determined by the ordinary differential equation

For ❬ ✓ =4 ✓ ✎ ● , one obtains from (1) the following exact solution to the original equation with arbitrary = ( ):

(2 − ) where ✖ 1 and ✖ 2 are arbitrary constants.

2 ❄ . The change of variable ❪ = ✳ 1− ▼ brings (1) to the generalized Emden–Fowler equation

A lot of exact solutions to equation (2) with various = ( ) can be found in Polyanin and Zaitsev (1995). ❳✬❨

Reference : V. F. Zaitsev and A. D. Polyanin (1996).

For ❍ ❜ ≠ 0, there are exact solutions of the form ✎ ✎ ✣

where the function = ( ✳ ) is determined by the ordinary differential equation

The change of variable 2 = brings (1) to the generalized Emden–Fowler equation

whose solutions with

+ ❡ ( −1 )=( ) and ( )=( ✹ + ❡ −2 ) ( ✹ , ❡ = const) can be found in Polyanin and Zaitsev (1995). ❳✬❨

Reference : V. F. Zaitsev and A. D. Polyanin (1996).

For ● ≠ 2 and ❜ ≠ 0, there are exact solutions of the form

where the function ✎ = (

) is determined by the ordinary differential equation

Exact solution in multiplicative form:

where ✏ = ✏ ( ☞ ) and ✑ = ✑ ( ✌ ) are determined by the ordinary differential equations ✓

[ ✗ ( ☞ ) ✏ ✧ ✢ ] ✧ ✢ =[ ✜ ln ✏ + ✕ 1 ( ☞ )+ ✖ ] ✏ , [ ( ✌ ) ✑ ✧ ❱ ] ✧ ❱ =[ ✜ ln ✑ + ✕ 2 ( ✌ )− ✖ ] ✑ ,

where ✖ is an arbitrary constant.

B.5.3-3. Other equations with two independent variables.

. Exact solution: r

, ✈ , ⑤ 1 , ⑤ 2 , and s 0 are arbitrary constants. r

2 q . Exact solution:

( s , t )= ⑥ ( s ) t 2 + ⑦ ( s ) t + ⑧ ( s ),

where the functions ⑥ = ⑥ ( s ), ⑦ = ⑦ ( s ), and ⑧ = ⑧ ( s ) are determined by the ordinary differential equations

The nonlinear equation (1) can be treated independently. For ③ ≡ const, its solution can be

expressed in terms of elliptic integrals. For 2 =

. Equations (2) and (3) can be solved successively (these are linear in the unknowns). Because

, a particular solution of (1) is

= ⑥ ( s ) is a particular solution of equation (2), the general solution of (2) is given by (see Polyanin and Zaitsev 1995)

( s )= ⑤ 1 ⑥ ( s )+ ⑤ 2 ⑥ ( s ) ✇

where ⑤ 1 and ⑤ 2 are arbitrary constants.

For ♦ ≠ 2 and ≠ 2, there is an exact solution of the form

2 1 ➄ = 2 ( ), = (2 − ) s + ❶ (2 − ❾ ) t 2− ➃ ♠ . Here, the function = ( ➀

) is determined by the ordinary differential equation r

Reference : V. F. Zaitsev and A. D. Polyanin (1996). 2 ➋

For ♦ ≠ 2 and ≠ 2, there is an exact solution of the form

For ➓ ➔ ➐ r ≠ 0, there is an exact solution of the form

rr

Here, the function ① = (

) is determined by the ordinary differential equation r

For ➓ ≠ 0, there is an exact solution of the form

≠ 0 and ❾ ≠ 2, there is an exact solution of the form r r

2 − ↔ ↕✛➝ 1 ➄ = 2 ( ), = s + ❶ (2 − ❾ ) ❷ .

Here, the function r = ( ➀ ) is determined by the ordinary differential equation r

Reference : V. F. Zaitsev and A. D. Polyanin (1996). 2 ➋

For ➓ ➔ ≠ 0 and ❾ ➐ ≠ 2, there is an exact solution of the form

2 − ↔ = ↕✛➝ ➆ ➁ ➔ s + ❶ (2 − ❾ ) ❷

B.5.3-4. Equations with three independent variables.

≠ 2, ❿ ≠ 2, and r ➤ ≠ 2, there is an exact solution of the form r

where the function r ( ➀ ) is determined by the ordinary differential equation r r

Reference : A. D. Polyanin and A. I. Zhurov (1998).

≠ 0, → ≠ 0, and ➳ ≠ 0, there is an exact solution of the form r r

2 + ➨ ➳ 2 ➟ , where the function ( ➀ ) is determined by the ordinary differential equation r r r

≠ 2, ❿ ≠ 2, and r ➳ ≠ 0, there is an exact solution of the form r

where the function ( r ➀ ) is determined by the ordinary differential equation r r

≠ 2, → ≠ 0, and ➳ ≠ 0, there is an exact solution of the form r r

where the function ( ➀ ) is determined by the ordinary differential equation r r r

Reference : A. D. Polyanin and A. I. Zhurov (1998).

B.5.4. Equations Containing Mixed Derivatives

B.5.4-1. Monge–Amp`ere equations.

. Exact solutions: r

where ⑤ 1 , ⑤ 2 , ⑤ 3 , ⑤ 4 , ⑤ 5 , and ⑤ 6 are arbitrary constants.

2 ③ q . Exact solutions for ( s ) > 0:

( s , t )= ➺ t ✇ ➻ ③ ( s ) ④ s + ⑥ ( s )+ ⑤ 1 t , where ⑥ ( s ) is an arbitrary function.

. Exact solution: r

( s , t )= ⑤ 1 t 2 − t ✇ ➼ ( s ) ④ s

where ⑤ 1 , ➽✛➽✛➽ , ⑤ 5 , and ❶ are arbitrary constants. r

2 q . Exact solution:

3 q . Exact solutions cubic in r t :

where ⑤ 1 , ➽✛➽✛➽ , ⑤ 5 , and ❶ are arbitrary constants.

4 q . See the solution of equation B.5.4.5 in Item 2 q with ➇ = 1.

. Exact solution quadratic in t : ( s , t )= ⑥ ( s ) t 2 + ➥➚⑤ 1 ✇ ⑥ 2 ( 1 s ) ④ s + ⑤ 2 ➩ t + ⑤ 2 ⑥ 1 3 ✇ ① ( s − ② ) ( ② ) ④ ② + ⑤ 3 s + ⑤ 4 .

The function ⑥ = ⑥ ( s ) is determined by the ordinary differential equation

2 ① . Exact solutions quartic in r t ① :

where ⑤ 1 , ➽✛➽✛➽ , ⑤ 5 , and ❶ are arbitrary constants.

3 q . See the solution of equation B.5.4.5 in Item 2 q with ➇ = 2.

Exact solution: ➓

( s , t )= ⑥ ( s ) t 2 + ⑦ ( s ) t + ⑧ ( s ),

where the functions ⑥ = ⑥ ( s ), ⑦ = ⑦ ( s ), and ⑧ = ⑧ ( s ) are determined by the system of ordinary differential equations

1 q . Exact solutions: r ➐ ➐ ➓ ➐

where ⑤ 1 , ➽✛➽✛➽ , ⑤ 5 , and ❶ are arbitrary constants. r

2 q . Exact solution:

( s t )= ⑥ ( s ) t ➘ , 2 .

The function ⑥ = ⑥ ( s ) is determined by the ordinary differential equation

Exact solution: ➐ r ➓

( +2 s ) t ➘ −

. Exact solutions:

( s , t )= ⑤ 1 ✇ ①

where ⑤ 1 , ⑤ 2 , ⑤ 3 , ⑤ 4 , ❶ , and ➔ are arbitrary constants. r

2 q . Exact solution:

)= ➙ ⑥ ( s ) exp ➣ , 1

where ⑥ = ⑥ ( s ) is determined by the ordinary differential equation

Exact solution: ➓

)= ⑥ ( s

st

where the function ⑥ = ⑥ ( s ) is determined by the ordinary differential equation

Exact solution: ➓

( s , t )= ⑤ 1 ✇ ①

where ⑤ 1 , ⑤ 2 , ⑤ 3 , ⑤ 4 , ❶ , and ➁ are arbitrary constants. B.5.4-2. Other equations with quadratic nonlinearities.

. Suppose ( s , t ) is a solution of the equation. Then the functions

where ⑥ ( s ) is an arbitrary function and ⑤ is an arbitrary constant, are also solutions of the equation.

2 q . Exact solutions: r

where ⑥ ( s ) is an arbitrary function and ⑤ 1 and r ⑤ 2 are arbitrary constants.

3 q . Exact solutions in implicit form:

where ⑥ ( ➬ ) and ⑦ ( ➮ ) are arbitrary functions and ➼ ( ➬ )= ✃ ❐ ( ➬ ) ❒ ➬ .

2 ( ➓ ) + ➪ ( ➓ ) ➊ + s( ➓ ).

Exact solution: ➓

where the functions ❮ ( ➱ ) and ❰ ( ➱ ) are determined by the system of ordinary differential equations

B.5.5. General Form Equations

B.5.5-1. Equations of the form 2

Exact solution: Ö

where ß and à are arbitrary constants and the function ❮ ( Þ ) is determined by the ordinary differential equation

Exact solution: Ö

where ß , à , ä , and å are arbitrary constants and the function ❮ ( æ ) is determined by the ordinary differential equation

Exact solution in multiplicative form: Ö

( Ô ,− å ) í Ôïî , where ß , à , and å are arbitrary constants.

, Ô )=[ ß cos( å Þ )+ à sin( å Þ )] ê 2 ( Ô

Exact solution: Ö

where ß , à , and å are arbitrary constants.

There are solutions of the form Ö

) cos( )+ ÷ ( ñ ) sin( Û ).

Exact solution: Ö

where ß , à , and å are arbitrary constants.

There are solutions of the form Ö

( Þ , Ô ) = cos( å Þ ) ú ( Ô ) + sin( å Þ ) û ( Ô ).

Exact solution in multiplicative form: Ö

, Ô )= ú ( Þ ) exp ë✝ì

where the function ú = ú ( Þ ) satisfies the linear ordinary differential equation ø ( Þ ) ú þ★þ Õ Õ = å ú .

1 Ö . Exact solution in multiplicative form:

, Ô )= ✂ exp ë➚å Þ + ì

where ✂ is an arbitrary constant.

2 ✁ . Exact solution in multiplicative form: Ý

where ß and à are arbitrary constants, and the function á â ú = ú ( Ô ) satisfies the ordinary differential equation ú ✄ þ = ú

3 ✁ . Exact solution in multiplicative form: Ý

( Þ , Ô )=[ ß sin( å Þ )+ à cos( å Þ )] ú ( Ô ),

where ß and à are arbitrary constants, and the function á â ú = ú ( Ô ) satisfies the ordinary differential

☎✝✆ equation = ,− å 2 ( ß 2 + à 2 ) ú ,− 2 ã .

Reference : Ph. W. Doyle (1996), the case ✞✠✟☛✡ ≡0 was considered.

Exact solution in multiplicative form: Ö

where ✂ 0 , ✂ 1 , and ✂ 2 are arbitrary constants, and the function á â ú = ú ( Ô ) satisfies the ordinary differential equation ú þ ✄

Reference : Ph. W. Doyle (1996), the case ✞✠✟☛✡ ≡0 was considered.

2 B.5.5-2. Equations of the form ✌

Exact solution in additive form: ñ

where the functions ú ( Þ ) and û ( Ô ) are determined by solving the second-order nonlinear ordinary differential equations ( ✂ is an arbitrary constant)

. Exact solution in multiplicative form:

where ß and à are arbitrary constants, and the function á â ú = ú ( Ô ) satisfies the ordinary differential equation ú ✄✒✄ þ★þ = ú

2 ✁ . Exact solution in multiplicative form:

( Þ , Ô )=[ ß sin( å Þ )+ à cos( å Þ )] ú ( Ô ),

where ß and à are arbitrary constants, and the function á â ú = ú ( Ô ) satisfies the ordinary differential

Exact solution in multiplicative form: ✔

where ✂ 0 , ✂ 1 , and ✂ 2 are arbitrary constants, and the function á â ú = ú ( Ô ) satisfies the ordinary differential equation ú þ★þ ✄✒✄ = ú

There are solutions of the form ✔

There are solutions of the form ✔

( Þ , Ô ) = cos( å Þ ) ú ( Ô ) + sin( å Þ ) û ( Ô ).