34.6.11 problem 11

Internal problem ID [6085]
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 : 11
Date solved : Monday, January 27, 2025 at 01:35:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 37

dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(4*x^2-9*a^2)/(4*a^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_2 \left (i x +a \right ) {\mathrm e}^{-\frac {i x}{a}}+\left (-i x +a \right ) c_1 \,{\mathrm e}^{\frac {i x}{a}}}{x^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 62

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(4*x^2-9*a^2)/(4*a^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left ((a c_2+c_1 x) \cos \left (\frac {x}{a}\right )+(c_2 x-a c_1) \sin \left (\frac {x}{a}\right )\right )}{x \sqrt {\frac {x}{a}}} \]