Internal problem ID [11101]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{x \lambda }-a \,{\mathrm e}^{2 x \lambda }\right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 43
dsolve(diff(y(x),x$2)+a*(lambda*exp(lambda*x)-a*exp(2*lambda*x))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}+c_{2} {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} \operatorname {Ei}_{1}\left (-\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) \]
✓ Solution by Mathematica
Time used: 1.32 (sec). Leaf size: 37
DSolve[y''[x]+a*(\[Lambda]*Exp[\[Lambda]*x]-a*Exp[2*\[Lambda]*x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {a e^{\lambda x}}{\lambda }} \left (c_2 \operatorname {ExpIntegralEi}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+c_1\right ) \]