61.29.32 problem 141

Internal problem ID [12641]
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 : 141
Date solved : Tuesday, January 28, 2025 at 08:03:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.314 (sec). Leaf size: 81

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

\[ y = x^{-a -\frac {1}{2}} {\mathrm e}^{\frac {x^{2}}{2}} \left (\operatorname {WhittakerM}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{1} +\operatorname {WhittakerW}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.345 (sec). Leaf size: 231

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

\[ y(x)\to i^{-a} (-1)^{\frac {1}{4} \left (1-\sqrt {4 a^2-4 a (-1)^n+1}\right )} x^{\frac {1}{2} \left (-\sqrt {4 a^2-4 a (-1)^n+1}-2 a+1\right )} \left (c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-2 a-2 n-\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),1-\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1},x^2\right )+c_2 i^{\sqrt {4 a^2-4 a (-1)^n+1}} x^{\sqrt {4 a^2-4 a (-1)^n+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )\right ) \]