60.1.455 problem 458

Internal problem ID [10469]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 458
Date solved : Monday, January 27, 2025 at 07:49:31 PM
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: 0.037 (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 &= \frac {c_{1} \sqrt {-a^{2}}-\ln \left (2\right )-\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )}{\sqrt {-a^{2}}} \\ y &= \frac {c_{1} \sqrt {-a^{2}}+\ln \left (2\right )+\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )}{\sqrt {-a^{2}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 120

DSolve[-1 + x^2*(-a^2 + x^2)*D[y[x],x]^2==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*}