61.30.32 problem 180

Internal problem ID [12680]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 180
Date solved : Tuesday, January 28, 2025 at 08:10:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 12.686 (sec). Leaf size: 1535

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

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 0.806 (sec). Leaf size: 98

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

\[ y(x)\to \frac {(k-x) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {-k^3+K[1] k^2+K[1]^2 k-K[1]^3+2 a K[1]^2+2 c+2 b K[1]}{(k-K[1]) (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right )}{k} \]