61.29.29 problem 138

Internal problem ID [12638]
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 : 138
Date solved : Tuesday, January 28, 2025 at 08:03:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 0.582 (sec). Leaf size: 63

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

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