Internal problem ID [4006]
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)]`]]
\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right )=0} \]
✓ Solution by Maple
Time used: 0.047 (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 \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \ln \left (-\frac {b}{2}-\frac {a}{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+\left (-b -a \right ) y \left (x \right )+a b}\right )}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}+\int _{}^{x}-\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}d \textit {\_a} +c_{1} = 0 \frac {\sqrt {\left (y \left (x \right )-a \right ) \left (y \left (x \right )-b \right )}\, \ln \left (-\frac {b}{2}-\frac {a}{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+\left (-b -a \right ) y \left (x \right )+a b}\right )}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}+\int _{}^{x}\frac {\sqrt {-f \left (\textit {\_a} \right ) \left (-y \left (x \right )+a \right ) \left (b -y \left (x \right )\right )}}{\sqrt {y \left (x \right )-a}\, \sqrt {y \left (x \right )-b}}d \textit {\_a} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 4.208 (sec). Leaf size: 89
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 ) y(x)\to a y(x)\to b \end{align*}