Internal problem ID [4008]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 27
Problem number: 767.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right ) \left (y-b \right ) \left (y-c \right )=0} \]
✓ Solution by Maple
Time used: 0.235 (sec). Leaf size: 158
dsolve(diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b)*(y(x)-c) = 0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )}}d \textit {\_a} +\int _{}^{x}-\frac {\sqrt {f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (-y \left (x \right )+a \right )}}{\sqrt {-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (-y \left (x \right )+a \right )}}d \textit {\_a} +c_{1} = 0 \int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )}}d \textit {\_a} +\int _{}^{x}\frac {\sqrt {f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (-y \left (x \right )+a \right )}}{\sqrt {-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (-y \left (x \right )+a \right )}}d \textit {\_a} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 39.221 (sec). Leaf size: 228
DSolve[(y'[x])^2+f[x](y[x]-a)(y[x]-b)(y[x]-c)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x-\sqrt {f(K[1])}dK[1]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x-\sqrt {f(K[1])}dK[1]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right ) y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x\sqrt {f(K[2])}dK[2]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x\sqrt {f(K[2])}dK[2]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right ) y(x)\to a y(x)\to b y(x)\to c \end{align*}