61.29.21 problem 130

Internal problem ID [12630]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 130
Date solved : Tuesday, January 28, 2025 at 03:23:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 21

dsolve(x^2*diff(y(x),x$2)-2*a*x*diff(y(x),x)+(-b^2*x^2+a*(a+1))*y(x)=0,y(x), singsol=all)
 

\[ y = x^{a} \left (c_{1} \sinh \left (b x \right )+c_{2} \cosh \left (b x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 35

DSolve[x^2*D[y[x],{x,2}]-2*a*x*D[y[x],x]+(-b^2*x^2+a*(a+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 x^a e^{-b x}+\frac {c_2 x^a e^{b x}}{2 b} \]