Equations Containing Exponential Functions and

1.4. Equations Containing Exponential Functions and

Arbitrary Parameters

1.4.1. Equations of the Form ÷ ø = ú ÷ 2 ø + ü 2 ( û , ù ) ø

This is a special case of equation 1.8.1.1 with Ý ( )= + ï .

1 ✝ . Particular solutions ( ✞ , ✟ , and ñ ✠ are arbitrary constants): ñ

. The substitution Þ ( , )= ☛ ( , ) exp ☞ ✡ ✄ ✆ Ý Ø + ï Ø leads to a constant coefficient equation,

= Õ Ü ✏ ✏ ☛ , which is considered in Subsection 1.1.1.

This is a special case of equation 1.8.1.6 with Ý é ( )= ï and ë ( )= ì☎✄ ✆ .

1 ✝ . Particular solutions ( ✞ ñ and ✠ are arbitrary constants): ñ ñ

2 ✝ . The transformation ñ

1 Õ 2 3 Õ 2 ( Ø , )= ☛ ( Ú , ) exp ó ï

leads to a constant coefficient equation, Õ Ü Ý✎☛ = Ü ß❦ß ✑☛ , which is considered in Subsection 1.1.1.

This is a special case of equation 1.8.1.2 with â

Particular solutions ( ✞ ñ and are arbitrary constants): ñ

( , )= ✞ exp(− ✠ ) ✓ ✔ ó ✡

, where ✓ ✔ ( ✜ ) and ✛ ✔ ( ✜ ) are the Bessel functions of the first and second kind, respectively.

exp(− ) ✛ ✔ ó ✡

This is a special case of equation 1.8.1.6 with ✤

( ★ )= ✩☎✄ ✆ ✪ and ✫ ( ★ )= ✬ .

1 ✝ . Particular solutions ( ✞ and ✠ are arbitrary constants):

2 ✹ . The transformation

2 2 ( Ø )= ✺ ( ✻ , ★ ) exp ✭ ✮

leads to a constant coefficient equation, ✼ ✪ ✺

= ✙ ✼ ✽✾✽✑✺ , which is considered in Subsection 1.1.1.

This is a special case of equation 1.8.1.3 with ✤

. Particular solutions ( ✳ and ✵ are arbitrary constants):

2 ✹ . The transformation

leads to a constant coefficient equation, ✼ ✪ ✺ = ❄ ✼ ✽✾✽✑✺ , which is considered in Subsection 1.1.1.

This is a special case of equation 1.8.7.5 with ✤

This is a special case of equation 1.8.1.1 with ✤

1 ✹ . Particular solutions ( ✳ , ● , and ❍ are arbitrary constants):

. The substitution ❃ ( , ★ )= ✺ ( , ★ ) exp ✭ ✮

leads to a constant coefficient equation,

= ❄ ✼ ✏ ✏ ✺ , which is considered in Subsection 1.1.1.

The substitution ✬ ( ,

)= Ø ✺ ( Ø , ★ ) exp ❏

leads to an equation of the form 1.4.1.3:

For ◗ = 0, see equation 1.4.1.1; for

= 0, see equation 1.4.1.3.

For ≠ 0, the transformation

where

leads to an equation of the form 1.4.1.3:

1.4.2. Equations of the Form ❲ ❳

1. ❖ P

= ➴ ❖ P 2 +( ❘ ❀ ❁ ❂ + ❙ ) ❖ P .

This is a special case of equation 1.8.2.1 with ◗

1 ✹ . Particular solutions ( ✳ , ● , and ✵ are arbitrary constants):

, we obtain a constant coefficient equation, ✼

. On passing from ▲ , to the new variables ▲ , ✻ = + ✮

= ❄ ✼ ✽✾✽ ❃ , which is considered in Subsection 1.1.1.

2. ❖ P

= ➴ ❖ P 2 +( ❘ ❀ ❁ ✒ + ❙ ) ❖ P .

This is a special case of equation 1.8.2.2 with ◗

1 ✹ . Particular solutions ( ✳ and ✵ are arbitrary constants):

2 2 where ✮ ❵ = ❵ ( ✵ ) is a root of the quadratic equation ❄ ❵ + ❱ ❵ + ✵ = 0; ❛ ( ❝ , ❍ ; ✻ ) and ❜ ( ❝ , ❍ ; ✻ ) are degenerate hypergeometric functions. [Regarding the degenerate hypergeometric functions, see

Supplement A.9 and the books by Abramowitz and Stegun (1964) and Bateman and Erd ´elyi (1953, Vol. 1)].

2 2 In solutions (1), the parameter ✮ ❵ can be considered arbitrary, and then ✵ =− ❄ ❵ − ❱ ❵ .

2 ✹ . Other particular solutions ( ✳ and ● are arbitrary constants):

3 ③ . The substitution ④ = s t ✉ leads to an equation of the form 1.3.5.4:

This is a special case of equation 1.8.2.3 with ⑩

( ✇ )= ⑥ s t ❹ + ⑦ .

1 ③ . Particular solutions ( ① , ❺ , and ❻ are arbitrary constants):

, ✇ )= ① ♥ ♠ ( ✇ )+ ❺ , ♠ ( ✇ ) = exp ♣ r s✑t ❹ + ⑦☎✇✲✈ ,

2 )= 2 ① exp ❼❴❻ ♥ ♠ ( ✇ )+ ❻ ❧ ♠ ( ✇ ) ♦ ✇✸❽ + ❺ .

2 ③ . On passing from ✇ , ♥ to the new variables ( ① is any number)

) ♦ ✇ + ① , ④ = ♥ ♠ ( ✇ ), where ♠ ( ✇ ) = exp ♣ r s t ❹ + ⑦☎✇✲✈ ,

for the function ❃ ( , ④

) we obtain a constant coefficient equation, ❃ = , which is considered in Subsection 1.1.1.

This is a special case of equation 1.8.2.1 with ⑩ ❸ ( ✇ )= ⑥

1 ③ . Particular solutions ( ① , ❺ , and ➃ are arbitrary constants):

. On passing from ✇ , ♥ to the new variables ✇ , ④ = ♥ + r s✑t ❹ + s ➂ ❹ , we obtain a constant coefficient

equation, ❃ = , which is considered in Subsection 1.1.1.

For ⑩ = 0, see equation 1.4.2.2; for ❻ = 0, see equation 1.4.2.1. For ❻ ≠ 0, the substitution ④ = ♥

+( ➅ ❻ ) ✇ leads to an equation of the form 1.4.2.2:

This is a special case of equation 1.8.1.6 with ⑩ ❸ ( ✇ )= ⑦ and ➇ ( ✇ )= ⑥

This is a special case of equation 1.8.1.6 with ⑩ ❸ ( ✇ )= ⑥

This is a special case of equation 1.8.1.6 with ⑩ ❸ ( ✇ )= ⑥

and ➇ ( ✇ )= ⑦ s ➂ ❹ .

This is a special case of equation 1.8.2.3 with ⑩

( ✇ )= ⑥ s t ❹ + ⑦ s ➂ ❹ .

1 ③ . Particular solutions ( ① , ❺ , and ➃ are arbitrary constants):

2 ③ . On passing from ✇ , ♥ to the new variables ( ① is any number)

( ✇ ) ♦ ✇ + ① , ④ = ♥ ♠ ( ✇ ), where ♠ ( ✇ ) = exp ♣ r s✑t ❹ + ❻ s ➂ ❹ ✈ ,

for the function ❃ ( , ④ ) we obtain a constant coefficient equation, = , which is considered in Subsection 1.1.1.

This is a special case of equation 1.8.2.6 with ⑩

( ✇ )= ⑦ s t ❹ .

1.4.3. Equations of the Form ➉ ➊ = ➌ ➉ ➊

= ➆ ⑧ ⑨ 2 + ❶ ⑧ ⑨ +( ❷ ❀ ❁ ❂ + s) ⑨ .

The substitution ⑩

( ♥ , ✇ )= ➐ ( ♥ , ✇ ) exp ❼ − q ♥ + r s✑t ❹ + ♣ ➑ − q ✈ ✇✎❽

leads to a constant coefficient equation, q

, which is considered in Subsection 1.1.1.

= ➆ ⑧ ⑨ 2 + ❶ ⑧ ⑨ +( ❷ ❀ ❁ ➄ + s) .

The substitution ⑩ (

, ✇ )= ➐ ( ♥ , ✇ ) exp ➒ − 2 ➓ ➔ ♥ → leads to an equation of the form 1.4.1.3:

The substitution ⑩ ➜

( ♥ , ✇ )= ➐ ( ♥ , ✇ ) exp ♣ − q ♥ + r s✑t ❹ +

4 leads to a constant coefficient equation, ➐

, which is considered in Subsection 1.1.1.

= ➆ ⑧ ⑨ ➜ 2 + ❶ ⑧ ⑨ +( ❷ ↔ ↕ ➄ +s ↔ ➛ ➙ ) .

The substitution ⑩ (

, ✇ )= ➐ ( ♥ , ✇ ) exp ➒ ➞ s ➝ ❹ → leads to an equation of the form 1.4.3.2:

On passing from ⑩ ✇ , ♥ to the new variables ( ① and ❺ are any numbers)

+ ❺ , ④ = ① ➜ ♥ s◆t ❹ ,

for the function ❾ ( , ④ ) we obtain a constant coefficient equation of the form 1.1.3:

The substitution ⑩

= s t ✉ leads to an equation of the form 1.3.5.4: ➜ ➜

qr

1.4.4. Equations of the Form ➉ ➊ = ➌ ➍ ➡ ➉ ➊ + ➎

This is a special case of equation 1.8.8.1 with q ❸ ( ✇ )= , ➇ ( ✇ ) = 0, ➢ ( ✇ )= ⑦ , and ➑ ( ✇ )= ⑥

st ❹

This is a special case of equation 1.8.8.1 with q ❸ ( ✇ )= , ➇ ( ✇ ) = 0, ➢ ( ✇ )= ⑥

This is a special case of equation 1.8.8.1 with q ❸ ( ✇ )= , ➇ ( ✇ )= ⑥ , ➢ ( ✇ ) = 0, and ➑ ( ✇ )= ⑦

This is a special case of equation 1.8.8.2 with q ❸ ( ✇ )= , ➇ ( ✇ ) = 0, and ➢ ( ✇ )= ⑥

st ❹

This is a special case of equation 1.8.8.4 with q ❸ ( ✇ )= and ➇ ( ✇ )= ⑥

This is a special case of equation 1.8.4.5 with ⑩

( ✇ )= ⑥ s t ❹ + ⑦ .

This is a special case of equation 1.8.8.7 with q ❸ ( ✇ )= , ➇ ( ✇ )= ⑥

, and ➢ ( ✇ )= ⑦ s ➝ ❹ .

1.4.5. Equations of the Form 2 ➉ ➊

This is a special case of equation 1.8.6.1 with q ❸ ( ♥ )=

Particular solutions ( ① , ➜ ❺ , and ➃ are arbitrary constants):

where ➽ 0 ( ➾ ) and ➺ 0 ( ➾ ) are the Bessel functions.

. Particular solutions ( ➸ , ➱ , and ➃ are arbitrary constants):

2 + exp(− 1 ) ➽ 0 ➻ exp ➒ −

where ➽ 0 ( ➾ ) and ➺ 0 ( ➾ ) are the Bessel functions.

2 ➵ ➮ . The substitution (

, ✇ )= ✃ ➓❴❐✎Ï ( ➯ , ✇ ) leads to an equation of the form 1.4.5.1: Ð Ï ❐ = ✃ ❒ ❮ Ð ❮ ❮ Ï .

. Particular solutions ( ➸ , ➱ , ➃ are arbitrary constants):

2 ➮ . The substitution ➫ ➵ ( ➯ , ✇ ) = exp ➒ ➓ ✃ Ô ❐ + Õ☎✇

( ➯ , ✇ ) leads to an equation of the form 1.4.5.1:

This equation describes heat transfer in a quiescent medium (solid body) in the case where thermal ➷ diffusivity is an exponential function of the coordinate. The equation can be rewritten in the

divergence form

that is more customary for applications.

1 ➮ . Particular solutions ( ➸ , ➱ , Ù , and ➃ are arbitrary constants):

where ➽ 1 ( ç ) and ➺ 1 ( ç ) are the Bessel functions, å 1 ( ç ) and æ 1 ( ç ) are the modified Bessel functions.

2 ➮ . A solution containing an arbitrary function of the space variable:

) is any infinitely differentiable function. This solution satisfies the initial condition

3 ➮ . A solution containing an arbitrary function of time:

where ì ( ✇ ) is any infinitely differentiable function. If ( ✇ ) is a polynomial, then the series has finitely many terms.

4 ➮ . The transformation ( Ù 1 , Ù 2 , and Ù 3 are any numbers)

leads to the same equation, up to the notation,

− ❒ 5 ❮ ➮ . The substitution ç = ✃ leads to an equation of the form 1.3.4.1:

6 ➵ ➮ . A series solution of the original equation (under constant values of at the boundary and at the initial instant) can be found in Lykov (1967).

1 )= and ( ✇ ) = 0. The substitution ➾

This is a special case of equation 1.8.5.2 with ➫

(1− ✃ ) leads to a constant coefficient equation of the form 1.1.4 with ➭

This is a special case of equation 1.8.5.2 with ➷

and ì ( ✇ )= Õ✶✃ ô ❐ .

1.4.6. Other Equations

This is a special case of equation 1.8.7.3 with ì è ( ✇ )= ✃ ❒ ❐ , ( ✇ )= ❰☎✃ Ô ❐ , and õ ( ✇ )= Õ✶✃ ô ❐ .

This is a special case of equation 1.8.7.4 with ❒ ( ✇ )= ✃ ❐ , è ( ✇ )= ❰☎✃ Ô ❐ ì , ( ✇ ) = 0, õ ( ✇ )= Õ✶✃ ô ❐

, and ( ✇ ) = 0.

The transformation ➫

, ✇ ) = exp ➻ ÷ ✃ ô ❐ ➳ Ï ( ➯ , í ), í = ✃ Ô ❐ ➃

leads to an equation of the form 1.4.5.1: ❿