83.30.6 problem 6

Internal problem ID [19385]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (D) at page 109
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 01:36:28 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.074 (sec). Leaf size: 36

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

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