61.34.7 problem 7

Internal problem ID [12771]
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
Date solved : Tuesday, January 28, 2025 at 08:24:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 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 \end{align*}

✓ Solution by Maple

Time used: 2.699 (sec). Leaf size: 219

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)
 

\[ y = \frac {-\csc \left (\frac {\pi \left (n +3\right )}{n +2}\right ) c_{1} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}}\right ) \pi {\left (-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )}^{\frac {1}{2 n +4}} {\mathrm e}^{-\frac {\lambda x}{2}}+\Gamma \left (\frac {n +3}{n +2}\right )^{2} {\left (-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}\right )}^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n +2}}{\lambda ^{2} b^{2} \left (n +2\right )^{2}}}\right ) c_{2} \left (n +2\right ) \left (b \,{\mathrm e}^{\frac {\lambda x}{2}}+{\mathrm e}^{-\frac {\lambda x}{2}} c \right )}{\left (n +2\right ) \Gamma \left (\frac {n +3}{n +2}\right )} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],{x,2}]+(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