Internal problem ID [4007]
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)]`]]
\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )=0} \]
✓ Solution by Maple
Time used: 0.078 (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 \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\int _{}^{x}\frac {\sqrt {f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}}d \textit {\_a} +c_{1} = 0 \frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\int _{}^{x}-\frac {\sqrt {f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-b}}d \textit {\_a} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 60.157 (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*}