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: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x \lambda }+\lambda \right ) y^{\prime }-y \,{\mathrm e}^{2 x \lambda } a \lambda =0} \]
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 80
dsolve(diff(y(x),x$2)+(a*exp(2*lambda*x)+lambda)*diff(y(x),x)-a*lambda*exp(2*lambda*x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{-\lambda x} \lambda \right )+c_{2} \left (\sqrt {\pi }\, \left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{-\lambda x} \lambda \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{\lambda x} \sqrt {a}}{2 \sqrt {\lambda }}\right )+\sqrt {\lambda }\, \sqrt {a}\, \sqrt {2}\, {\mathrm e}^{-\frac {a \,{\mathrm e}^{2 \lambda x}}{2 \lambda }}\right ) \]
✓ Solution by Mathematica
Time used: 0.283 (sec). Leaf size: 129
DSolve[y''[x]+(a*Exp[2*\[Lambda]*x]+\[Lambda])*y'[x]-a*\[Lambda]*Exp[2*\[Lambda]*x]*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt {2 \pi } c_2 \left (a e^{2 \lambda x}+\lambda \right ) \text {erf}\left (\frac {\sqrt {a \lambda e^{2 \lambda x}}}{\sqrt {2} \lambda }\right )-4 i \sqrt {2} a c_1 e^{2 \lambda x}+2 c_2 e^{-\frac {a e^{2 \lambda x}}{2 \lambda }} \sqrt {a \lambda e^{2 \lambda x}}-4 i \sqrt {2} c_1 \lambda }{4 \sqrt {a \lambda e^{2 \lambda x}}} \]