83.41.6 problem 2 (v)

Internal problem ID [19486]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise at end of chapter VIII. Page 141
Problem number : 2 (v)
Date solved : Tuesday, January 28, 2025 at 01:46:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 41

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

\[ y \left (x \right ) = c_{1} \sin \left (\frac {\sqrt {a^{2}-x^{2}}}{\sqrt {a}}\right )+c_{2} \cos \left (\frac {\sqrt {a^{2}-x^{2}}}{\sqrt {a}}\right ) \]

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 69

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

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