Internal problem ID [3684]
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]
Solve \begin {gather*} \boxed {\left (1-a y\right ) \left (y^{\prime }\right )^{2}-a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 815
dsolve((1-a*y(x))*diff(y(x),x)^2 = a*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {\RootOf \left (4 a^{2} c_{1}^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}-4 \sqrt {a^{2}}\, c_{1} \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, c_{1} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} c_{1}^{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 \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, c_{1} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} c_{1}^{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 )+\RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}-4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, c_{1} \textit {\_Z} +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} c_{1}^{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 \relax (x ) = \frac {\RootOf \left (4 a^{2} c_{1}^{2}-8 a^{2} c_{1} x +4 a^{2} x^{2}+4 \sqrt {a^{2}}\, c_{1} \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, c_{1} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} c_{1}^{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 \RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, c_{1} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} c_{1}^{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 )+\RootOf \left (4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-8 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) c_{1} a^{2} x +4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) a^{2} x^{2}+4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, c_{1} \textit {\_Z} -4 \left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \sqrt {a^{2}}\, \textit {\_Z} x +\left (\tan ^{2}\left (\textit {\_Z} \right )\right ) \textit {\_Z}^{2}+4 a^{2} c_{1}^{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: 0.897 (sec). Leaf size: 151
DSolve[(1-a y[x]) (y'[x])^2==a y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}+\frac {a \log \left (\sqrt {1-\text {$\#$1} a}-\sqrt {\text {$\#$1}} \sqrt {-a}\right )}{(-a)^{3/2}}\&\right ]\left [-\sqrt {a} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\sqrt {\text {$\#$1}} \sqrt {1-\text {$\#$1} a}+\frac {a \log \left (\sqrt {1-\text {$\#$1} a}-\sqrt {\text {$\#$1}} \sqrt {-a}\right )}{(-a)^{3/2}}\&\right ]\left [\sqrt {a} x+c_1\right ] \\ y(x)\to 0 \\ \end{align*}