Internal problem ID [10964]
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 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 130.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (a +1\right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 23
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)
\[ y \left (x \right ) = c_{1} x^{a} \sinh \left (b x \right )+c_{2} x^{a} \cosh \left (b x \right ) \]
✓ Solution by Mathematica
Time used: 0.08 (sec). Leaf size: 35
DSolve[x^2*y''[x]-2*a*x*y'[x]+(-b^2*x^2+a*(a+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 x^a e^{-b x}+\frac {c_2 x^a e^{b x}}{2 b} \]