34.6.9 problem 9

Internal problem ID [6083]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number : 9
Date solved : Monday, January 27, 2025 at 01:35:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+n^{2} y&=\frac {6 y}{x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 53

dsolve(diff(y(x),x$2)+n^2*y(x)=6*y(x)/x^2,y(x), singsol=all)
 

\[ y = \frac {\left (c_1 \,n^{2} x^{2}+3 c_2 n x -3 c_1 \right ) \cos \left (n x \right )+\sin \left (n x \right ) \left (c_2 \,n^{2} x^{2}-3 c_1 n x -3 c_2 \right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 79

DSolve[D[y[x],{x,2}]+n^2*y[x]==6*y[x]/x^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (c_2 \left (-n^2\right ) x^2+3 c_1 n x+3 c_2\right ) \cos (n x)+\left (c_1 \left (n^2 x^2-3\right )+3 c_2 n x\right ) \sin (n x)\right )}{(n x)^{5/2}} \]