61.34.27 problem 27
Internal
problem
ID
[12791]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.3-1.
Equations
with
exponential
functions
Problem
number
:
27
Date
solved
:
Tuesday, January 28, 2025 at 08:24:36 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 1.237 (sec). Leaf size: 141
dsolve(diff(y(x),x$2)+(a*exp(lambda*x)+b)*diff(y(x),x)+( alpha*exp(2*lambda*x)+ beta*exp(lambda*x) + gamma )*y(x)=0,y(x), singsol=all)
\[
y = {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -\lambda x \left (b +\lambda \right )}{2 \lambda }} \left (\operatorname {WhittakerM}\left (-\frac {\left (b +\lambda \right ) a -2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {\left (b +\lambda \right ) a -2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right )
\]
✓ Solution by Mathematica
Time used: 1.812 (sec). Leaf size: 250
DSolve[D[y[x],{x,2}]+(a*Exp[\[Lambda]*x]+b)*D[y[x],x]+( alpha*Exp[2*\[Lambda]*x]+ \[Beta]*Exp[\[Lambda]*x] + \[Gamma] )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {-2 \beta +a (b+\lambda )+\sqrt {a^2-4 \alpha } \left (\lambda +\sqrt {b^2-4 \gamma }\right )}{2 \sqrt {a^2-4 \alpha } \lambda },\frac {\lambda +\sqrt {b^2-4 \gamma }}{\lambda },\frac {\sqrt {a^2-4 \alpha } e^{x \lambda }}{\lambda }\right )+c_2 L_{\frac {2 \beta -a (b+\lambda )-\sqrt {a^2-4 \alpha } \left (\lambda +\sqrt {b^2-4 \gamma }\right )}{2 \sqrt {a^2-4 \alpha } \lambda }}^{\frac {\sqrt {b^2-4 \gamma }}{\lambda }}\left (\frac {\sqrt {a^2-4 \alpha } e^{x \lambda }}{\lambda }\right )\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {b+a K[1]+\sqrt {a^2-4 \alpha } K[1]-\sqrt {b^2-4 \gamma }}{2 \lambda K[1]}dK[1]\right )
\]