Internal problem ID [4009]
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)]`]]
\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )=0} \]
✓ Solution by Maple
Time used: 0.078 (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 {c b -c y \left (x \right )-b y \left (x \right )+y \left (x \right )^{2}}+2 a y \left (x \right )-b y \left (x \right )-c y \left (x \right )-a b -a c +2 c b}{y \left (x \right )-a}\right ) \sqrt {a^{2}-a b -a c +c b}\, \sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-c}}{\left (b -a \right ) \left (a -c \right ) \sqrt {c b -c y \left (x \right )-b y \left (x \right )+y \left (x \right )^{2}}}+\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-c}\, \sqrt {y \left (x \right )-b}}d \textit {\_a} +c_{1} = 0 \frac {\ln \left (\frac {2 \sqrt {a^{2}-a b -a c +c b}\, \sqrt {c b -c y \left (x \right )-b y \left (x \right )+y \left (x \right )^{2}}+2 a y \left (x \right )-b y \left (x \right )-c y \left (x \right )-a b -a c +2 c b}{y \left (x \right )-a}\right ) \sqrt {a^{2}-a b -a c +c b}\, \sqrt {y \left (x \right )-b}\, \sqrt {y \left (x \right )-c}}{\left (b -a \right ) \left (a -c \right ) \sqrt {c b -c y \left (x \right )-b y \left (x \right )+y \left (x \right )^{2}}}+\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-c}\, \sqrt {y \left (x \right )-b}}d \textit {\_a} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 60.323 (sec). Leaf size: 251
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 \frac {c (a-b)+b (a-c) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \sqrt {c-a} \left (\int _1^x-i \sqrt {f(K[1])}dK[1]+c_1\right )\right )}{(a-c) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \sqrt {c-a} \left (\int _1^x-i \sqrt {f(K[1])}dK[1]+c_1\right )\right )+a-b} y(x)\to \frac {c (a-b)+b (a-c) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \sqrt {c-a} \left (\int _1^xi \sqrt {f(K[2])}dK[2]+c_1\right )\right )}{(a-c) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \sqrt {c-a} \left (\int _1^xi \sqrt {f(K[2])}dK[2]+c_1\right )\right )+a-b} \end{align*}