Internal problem ID [4192]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 958.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
\[ \boxed {\left (1-a y\right ) {y^{\prime }}^{2}-a y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 815
dsolve((1-a*y(x))*diff(y(x),x)^2 = a*y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 y \left (x \right ) = \frac {\operatorname {RootOf}\left (4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, c_{1} \operatorname {RootOf}\left (4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} a^{2} x +4 \tan \left (\textit {\_Z} \right )^{2} a^{2} x^{2}-4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} c_{1} \textit {\_Z} +4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} x +\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}+4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, c_{1} \textit {\_Z} +4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )+4 \sqrt {a^{2}}\, x \operatorname {RootOf}\left (4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} a^{2} x +4 \tan \left (\textit {\_Z} \right )^{2} a^{2} x^{2}-4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} c_{1} \textit {\_Z} +4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} x +\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}+4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, c_{1} \textit {\_Z} +4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )+\operatorname {RootOf}\left (4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} a^{2} x +4 \tan \left (\textit {\_Z} \right )^{2} a^{2} x^{2}-4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} c_{1} \textit {\_Z} +4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} x +\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}+4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, c_{1} \textit {\_Z} +4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )^{2}+\textit {\_Z}^{2}-2 \textit {\_Z} \right )}{2 a} y \left (x \right ) = \frac {\operatorname {RootOf}\left (4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, c_{1} \operatorname {RootOf}\left (4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} a^{2} x +4 \tan \left (\textit {\_Z} \right )^{2} a^{2} x^{2}+4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} c_{1} \textit {\_Z} -4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} x +\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}+4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, c_{1} \textit {\_Z} -4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )-4 \sqrt {a^{2}}\, x \operatorname {RootOf}\left (4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} a^{2} x +4 \tan \left (\textit {\_Z} \right )^{2} a^{2} x^{2}+4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} c_{1} \textit {\_Z} -4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} x +\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}+4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, c_{1} \textit {\_Z} -4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )+\operatorname {RootOf}\left (4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} a^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} a^{2} x +4 \tan \left (\textit {\_Z} \right )^{2} a^{2} x^{2}+4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} c_{1} \textit {\_Z} -4 \sqrt {a^{2}}\, \tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} x +\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}+4 c_{1}^{2} a^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, c_{1} \textit {\_Z} -4 \sqrt {a^{2}}\, x \textit {\_Z} +\textit {\_Z}^{2}-1\right )^{2}+\textit {\_Z}^{2}-2 \textit {\_Z} \right )}{2 a} \end{align*}
✓ Solution by Mathematica
Time used: 0.571 (sec). Leaf size: 147
DSolve[(1-a y[x]) (y'[x])^2==a y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {2 \arctan \left (\frac {\sqrt {\text {$\#$1}} \sqrt {a}}{\sqrt {1-\text {$\#$1} a}-1}\right )}{\sqrt {a}}+\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}\&\right ]\left [-\sqrt {a} x+c_1\right ] y(x)\to \text {InverseFunction}\left [\frac {2 \arctan \left (\frac {\sqrt {\text {$\#$1}} \sqrt {a}}{\sqrt {1-\text {$\#$1} a}-1}\right )}{\sqrt {a}}+\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}\&\right ]\left [\sqrt {a} x+c_1\right ] y(x)\to 0 \end{align*}