60.1.489 problem 492

Internal problem ID [10503]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 492
Date solved : Monday, January 27, 2025 at 08:24:21 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 115

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

\begin{align*} y &= 0 \\ a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y^{2}}+a \right )}{y}\right )-\sqrt {a^{2}-y^{2}}-c_{1} +x &= 0 \\ -a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y^{2}}+a \right )}{y}\right )+\sqrt {a^{2}-y^{2}}-c_{1} +x &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.334 (sec). Leaf size: 102

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}