77.1.136 problem 163 (page 240)

Internal problem ID [18026]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 163 (page 240)
Date solved : Tuesday, January 28, 2025 at 11:22:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 19

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

\[ y = c_{1} x^{-n -1}+c_{2} x^{n} \]

✓ Solution by Mathematica

Time used: 0.056 (sec). Leaf size: 84

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

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