61.34.26 problem 26

Internal problem ID [12790]
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 : 26
Date solved : Tuesday, January 28, 2025 at 08:24:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.267 (sec). Leaf size: 114

dsolve(diff(y(x),x$2)+(a*exp(x)+b)*diff(y(x),x)+( c*(a-c)*exp(2*x)+ (a*k+b*c+c-2*c*k)*exp(x) + k*(b-k) )*y(x)=0,y(x), singsol=all)
 

\[ y = -\left (a -2 c \right )^{2 k -b} {\mathrm e}^{-\frac {b x}{2}-\frac {{\mathrm e}^{x} a}{2}} \operatorname {WhittakerM}\left (k -\frac {b}{2}, k -\frac {b}{2}+\frac {1}{2}, \left (a -2 c \right ) {\mathrm e}^{x}\right ) c_{2} +\left (\left (a -2 c \right ) {\mathrm e}^{x}\right )^{k -\frac {b}{2}} c_{2} \left (a -2 c \right )^{2 k -b} \left (-1-2 k +b \right ) {\mathrm e}^{\left (c -a \right ) {\mathrm e}^{x}-\frac {b x}{2}}+c_{1} {\mathrm e}^{-k x -{\mathrm e}^{x} c} \]

✓ Solution by Mathematica

Time used: 1.931 (sec). Leaf size: 71

DSolve[D[y[x],{x,2}]+(a*Exp[x]+b)*D[y[x],x]+( c*(a-c)*Exp[2*x]+ (a*k+b*c+c-2*c*k)*Exp[x] + k*(b-k) )*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (e^x\right )^{-k} e^{c \left (-e^x\right )-k} \left (c_2 \int _1^{e^x}\exp \left (\int _1^{K[2]}-\frac {b-2 k+a K[1]-2 c K[1]+1}{K[1]}dK[1]\right )dK[2]+c_1\right ) \]