Internal problem ID [11116]
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: 28.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+{\mathrm e}^{2 \mu x} b +{\mathrm e}^{\mu x} c +k \right ) y=0} \]
✗ Solution by Maple
dsolve(diff(y(x),x$2)+(2*a*exp(lambda*x)-lambda)*diff(y(x),x)+( a^2*exp(2*lambda*x) + b*exp(2*mu*x) + c*exp(mu*x) + k )*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 2.29 (sec). Leaf size: 290
DSolve[y''[x]+(2*a*Exp[\[Lambda]*x]-\[Lambda])*y'[x]+( a^2*Exp[2*\[Lambda]*x] + b*Exp[2*\[Mu]*x] + c*Exp[\[Mu]*x] + k )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (\left (e^x\right )^{\lambda }\right )^{\frac {\lambda -1}{2 \lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\mu }{2}} 2^{\frac {\sqrt {\mu ^2 \left (\lambda ^2-4 k\right )}+\mu ^2}{2 \mu ^2}} \left (\left (e^x\right )^{\mu }\right )^{\frac {\sqrt {\mu ^2 \left (\lambda ^2-4 k\right )}+\mu ^2}{2 \mu ^2}} e^{-\frac {a \left (e^x\right )^{\lambda }}{\lambda }+\frac {i \sqrt {b} \left (e^x\right )^{\mu }}{\mu }} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\mu ^2-\frac {i c \mu }{\sqrt {b}}+\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{2 \mu ^2},\frac {\mu ^2+\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{\mu ^2},-\frac {2 i \sqrt {b} \left (e^x\right )^{\mu }}{\mu }\right )+c_2 L_{\frac {i c}{2 \sqrt {b} \mu }-\frac {\mu ^2+\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{2 \mu ^2}}^{\frac {\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{\mu ^2}}\left (-\frac {2 i \sqrt {b} \left (e^x\right )^{\mu }}{\mu }\right )\right ) \]