Internal problem ID [11095]
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: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.844 (sec). Leaf size: 218
dsolve(diff(y(x),x$2)+(a*exp(2*lambda*x)*(b*exp(lambda*x)+c)^n-1/4*lambda^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {x \lambda }{2}} \Gamma \left (\frac {n +1}{n +2}\right )^{2} {\left (-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )}^{\frac {1}{2 n +4}} c_{1} \left (n +2\right ) \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}}\right )+\csc \left (\frac {\pi \left (n +1\right )}{n +2}\right ) {\left (-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )}^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{x \lambda }+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}}\right ) \pi c_{2} \left (b \,{\mathrm e}^{\frac {x \lambda }{2}}+{\mathrm e}^{-\frac {x \lambda }{2}} c \right )}{\left (n +2\right ) \Gamma \left (\frac {n +1}{n +2}\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+(a*Exp[2*\[Lambda]*x]*(b*Exp[\[Lambda]*x]+c)^n-1/4*\[Lambda]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved