Internal problem ID [3498]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 765.
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 ) \left (y-b \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 212
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b) = 0,y(x), singsol=all)
\begin{align*} \frac {\sqrt {\left (y \relax (x )-a \right ) \left (y \relax (x )-b \right )}\, \ln \left (-\frac {a}{2}-\frac {b}{2}+y \relax (x )+\sqrt {y \relax (x )^{2}+\left (-a -b \right ) y \relax (x )+a b}\right )}{\sqrt {y \relax (x )-a}\, \sqrt {y \relax (x )-b}}+\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \relax (x )+a \right ) \left (-y \relax (x )+b \right )}}{\sqrt {y \relax (x )-a}\, \sqrt {y \relax (x )-b}}d \textit {\_a} +c_{1} = 0 \\ \frac {\sqrt {\left (y \relax (x )-a \right ) \left (y \relax (x )-b \right )}\, \ln \left (-\frac {a}{2}-\frac {b}{2}+y \relax (x )+\sqrt {y \relax (x )^{2}+\left (-a -b \right ) y \relax (x )+a b}\right )}{\sqrt {y \relax (x )-a}\, \sqrt {y \relax (x )-b}}+\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \relax (x )+a \right ) \left (-y \relax (x )+b \right )}}{\sqrt {y \relax (x )-a}\, \sqrt {y \relax (x )-b}}d \textit {\_a} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.159 (sec). Leaf size: 79
DSolve[(y'[x])^2+ f[x](y[x]-a)(y[x]-b)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left ((b-a) \cosh \left (\int _1^x-i \sqrt {f(K[2])}dK[2]+c_1\right )+a+b\right ) \\ y(x)\to \frac {1}{2} \left ((b-a) \cosh \left (\int _1^xi \sqrt {f(K[3])}dK[3]+c_1\right )+a+b\right ) \\ \end{align*}