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