29.32.2 problem 936

Internal problem ID [5514]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 32
Problem number : 936
Date solved : Monday, January 27, 2025 at 11:34:53 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 111

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

\begin{align*} y \left (x \right ) &= \frac {c_{1} \sqrt {-a^{2}}-\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )-\ln \left (2\right )}{\sqrt {-a^{2}}} \\ y \left (x \right ) &= \frac {c_{1} \sqrt {-a^{2}}+\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )+\ln \left (2\right )}{\sqrt {-a^{2}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 120

DSolve[x^2(a^2-x^2) (D[y[x],x])^2+1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {x \sqrt {x^2-a^2} \arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \\ y(x)\to \frac {x \sqrt {x^2-a^2} \arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \\ \end{align*}