Internal problem ID [11102]
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: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)+(a+b)*exp(lambda*x)*diff(y(x),x)+a*exp(lambda*x)*(b*exp(lambda*x)+lambda)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,{\mathrm e}^{x \lambda }}{\lambda }} \left (c_{1} +\operatorname {expIntegral}_{1}\left (-\frac {{\mathrm e}^{x \lambda } \left (a -b \right )}{\lambda }\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 2.377 (sec). Leaf size: 40
DSolve[y''[x]+(a+b)*Exp[\[Lambda]*x]*y'[x]+a*Exp[\[Lambda]*x]*(b*Exp[\[Lambda]*x]+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }} \left (c_2 \operatorname {ExpIntegralEi}\left (\frac {(a-b) e^{x \lambda }}{\lambda }\right )+c_1\right ) \]