Internal problem ID [11112]
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: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 54
dsolve(diff(y(x),x$2)+(2*a*exp(lambda*x)+b)*diff(y(x),x)+(a^2*exp(2*lambda*x)+a*(b+lambda)*exp(lambda*x)+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {b \lambda x +2 \,{\mathrm e}^{x \lambda } a}{2 \lambda }} \left (c_{1} \sinh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )+c_{2} \cosh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.589 (sec). Leaf size: 82
DSolve[y''[x]+(2*a*Exp[\[Lambda]*x]+b)*y'[x]+(a^2*Exp[2*\[Lambda]*x]+a*(b+\[Lambda])*Exp[\[Lambda]*x]+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (c_2 e^{x \sqrt {b^2-4 c}}+c_1 \sqrt {b^2-4 c}\right ) e^{-\frac {a e^{\lambda x}}{\lambda }-\frac {1}{2} x \left (\sqrt {b^2-4 c}+b\right )}}{\sqrt {b^2-4 c}} \]