Internal problem ID [3499]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 27
Problem number: 766.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+f \relax (x ) \left (y-a \right )^{2} \left (y-b \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 110
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)^2*(y(x)-b) = 0,y(x), singsol=all)
\begin{align*} \frac {2 \arctan \left (\frac {\sqrt {y \relax (x )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\int _{}^{x}\frac {\sqrt {f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )}}{\sqrt {y \relax (x )-b}}d \textit {\_a} +c_{1} = 0 \\ \frac {2 \arctan \left (\frac {\sqrt {y \relax (x )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\int _{}^{x}-\frac {\sqrt {f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )}}{\sqrt {y \relax (x )-b}}d \textit {\_a} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.132 (sec). Leaf size: 93
DSolve[(y'[x])^2+f[x](y[x]-a)^2 (y[x]-b)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x-\sqrt {f(K[1])}dK[1]+c_1\right )\right ) \\ y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x\sqrt {f(K[2])}dK[2]+c_1\right )\right ) \\ \end{align*}