Internal problem ID [3501]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 27
Problem number: 768.
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 ) \left (y-c \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.066 (sec). Leaf size: 378
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)^2*(y(x)-b)*(y(x)-c) = 0,y(x), singsol=all)
\begin{align*} \frac {\ln \left (\frac {2 \sqrt {a^{2}-a b -a c +c b}\, \sqrt {y \relax (x )^{2}-b y \relax (x )-c y \relax (x )+c b}+2 a y \relax (x )-b y \relax (x )-c y \relax (x )-a b -a c +2 c b}{y \relax (x )-a}\right ) \sqrt {a^{2}-a b -a c +c b}\, \sqrt {y \relax (x )-b}\, \sqrt {y \relax (x )-c}}{\left (c -a \right ) \left (a -b \right ) \sqrt {y \relax (x )^{2}-b y \relax (x )-c y \relax (x )+c b}}+\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \relax (x )+c \right ) \left (-y \relax (x )+b \right )}}{\sqrt {y \relax (x )-c}\, \sqrt {y \relax (x )-b}}d \textit {\_a} +c_{1} = 0 \\ \frac {\ln \left (\frac {2 \sqrt {a^{2}-a b -a c +c b}\, \sqrt {y \relax (x )^{2}-b y \relax (x )-c y \relax (x )+c b}+2 a y \relax (x )-b y \relax (x )-c y \relax (x )-a b -a c +2 c b}{y \relax (x )-a}\right ) \sqrt {a^{2}-a b -a c +c b}\, \sqrt {y \relax (x )-b}\, \sqrt {y \relax (x )-c}}{\left (c -a \right ) \left (a -b \right ) \sqrt {y \relax (x )^{2}-b y \relax (x )-c y \relax (x )+c b}}+\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \relax (x )+c \right ) \left (-y \relax (x )+b \right )}}{\sqrt {y \relax (x )-c}\, \sqrt {y \relax (x )-b}}d \textit {\_a} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 155
DSolve[(y'[x])^2+f[x](y[x]-a)^2 (y[x]-b) (y[x]-c)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a-\frac {2 (a-b) (a-c)}{(c-b) \cos \left (\sqrt {a-b} \sqrt {c-a} \left (\int _1^x-i \sqrt {f(K[1])}dK[1]+c_1\right )\right )+2 a-b-c} \\ y(x)\to a-\frac {2 (a-b) (a-c)}{(c-b) \cos \left (\sqrt {a-b} \sqrt {c-a} \left (\int _1^xi \sqrt {f(K[2])}dK[2]+c_1\right )\right )+2 a-b-c} \\ \end{align*}