29.8 problem 117

Internal problem ID [10951]

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: 117.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

\[ \boxed {x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y=0} \]

Solution by Maple

Time used: 0.0 (sec). Leaf size: 50

dsolve(x^2*diff(y(x),x$2)-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y(x)=0,y(x), singsol=all)
 

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

Solution by Mathematica

Time used: 0.45 (sec). Leaf size: 66

DSolve[x^2*y''[x]-(a^2*x^4+a*(2*b-1)*x^2+b*(b+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-\frac {a x^2}{2}} x^{-b} \left (a c_2 x^{2 b+3} \left (-a x^2\right )^{-b-\frac {3}{2}} \Gamma \left (b+\frac {1}{2},-a x^2\right )+2 c_1\right ) \]