Internal problem ID [11110]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 x \lambda } \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 111
dsolve(diff(y(x),x$2)-diff(y(x),x)+(a*exp(2*lambda*x)*(b*exp(lambda*x)+c)^n+1/4-1/4*lambda^2 )*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\left (\lambda -1\right ) x}{2}} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1+n}{n +2}\right ], -\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )+c_{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {n +3}{n +2}\right ], -\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right ) \left ({\mathrm e}^{\frac {x \left (\lambda +1\right )}{2}} b +{\mathrm e}^{-\frac {\left (\lambda -1\right ) x}{2}} c \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]-y'[x]+(a*Exp[2*\[Lambda]*x]*(b*Exp[\[Lambda]*x]+c)^n+1/4-1/4*\[Lambda]^2 )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved