33.20 problem 982

Internal problem ID [3707]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 33
Problem number: 982.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [_quadrature]

Solve \begin {gather*} \boxed {\left (a^{2}-y^{2}\right ) \left (y^{\prime }\right )^{2}-y^{2}=0} \end {gather*}

Solution by Maple

Time used: 0.138 (sec). Leaf size: 126

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

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

Solution by Mathematica

Time used: 0.293 (sec). Leaf size: 102

DSolve[(a^2-y[x]^2) (y'[x])^2==y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

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