32.19 problem 953

Internal problem ID [4187]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 32
Problem number: 953.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [_quadrature]

\[ \boxed {y {y^{\prime }}^{2}+y=a} \]

✓ Solution by Maple

Time used: 0.078 (sec). Leaf size: 1583

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

\begin{align*} y \left (x \right ) = a y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a}{2}+\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} -\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a -4 c_{1} +4 x \right )}{2}+\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} -\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a}{2}-\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} +\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a +4 c_{1} -4 x \right )}{2}-\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} +\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a x \textit {\_Z} +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} \end{align*}

✓ Solution by Mathematica

Time used: 0.429 (sec). Leaf size: 106

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

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