Internal problem ID [3679]
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]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}+y-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.127 (sec). Leaf size: 827
dsolve(y(x)*diff(y(x),x)^2+y(x) = a,y(x), singsol=all)
\begin{align*} y \relax (x ) = a \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a -2 c_{1}+2 x \right )}{2}+\frac {a}{2} \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}+4 c_{1} \textit {\_Z} a -4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a +2 c_{1}-2 x \right )}{2}+\frac {a}{2} \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (-\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a +2 c_{1}-2 x \right )}{2}+\frac {a}{2} \\ y \relax (x ) = \frac {\tan \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (\RootOf \left (\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} \textit {\_Z}^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a x \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) x^{2}+a^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} a +4 a \textit {\_Z} x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a -2 c_{1}+2 x \right )}{2}+\frac {a}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.315 (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 \text {ArcTan}\left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {a-\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {a-\text {$\#$1}}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [a \text {ArcTan}\left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {a-\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {a-\text {$\#$1}}\&\right ][x+c_1] \\ y(x)\to a \\ \end{align*}