Internal problem ID [4230]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 998.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+a y^{2}=b \,x^{2}+a b} \]
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 923
dsolve((a-b)*y(x)^2*diff(y(x),x)^2-2*b*x*y(x)*diff(y(x),x)-a*b-b*x^2+a*y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\sqrt {b \left (x^{2}+a -b \right ) \left (a -b \right )}}{a -b} y \left (x \right ) = -\frac {\sqrt {b \left (x^{2}+a -b \right ) \left (a -b \right )}}{a -b} \int _{\textit {\_b}}^{x}-\frac {\textit {\_a} b +\sqrt {a \left (\textit {\_a}^{2} b -a y \left (x \right )^{2}+b y \left (x \right )^{2}+a b -b^{2}\right )}}{-a y \left (x \right )^{2}+\textit {\_a}^{2} b +b y \left (x \right )^{2}+\sqrt {a \left (\textit {\_a}^{2} b -a y \left (x \right )^{2}+b y \left (x \right )^{2}+a b -b^{2}\right )}\, \textit {\_a} +a b -b^{2}}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {\textit {\_f} \left (a -b \right )}{-a \,\textit {\_f}^{2}+b \,x^{2}+b \,\textit {\_f}^{2}+\sqrt {a \left (-a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+b \,x^{2}+a b -b^{2}\right )}\, x +a b -b^{2}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {\left (\textit {\_a} b +\sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}\right ) \left (-2 a \textit {\_f} +2 \textit {\_f} b +\frac {\textit {\_a} a \left (-2 a \textit {\_f} +2 \textit {\_f} b \right )}{2 \sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}}\right )}{\left (-a \,\textit {\_f}^{2}+\textit {\_a}^{2} b +b \,\textit {\_f}^{2}+\sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}\, \textit {\_a} +a b -b^{2}\right )^{2}}-\frac {a \left (-2 a \textit {\_f} +2 \textit {\_f} b \right )}{2 \left (-a \,\textit {\_f}^{2}+\textit {\_a}^{2} b +b \,\textit {\_f}^{2}+\sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}\, \textit {\_a} +a b -b^{2}\right ) \sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \int _{\textit {\_b}}^{x}-\frac {-\textit {\_a} b +\sqrt {a \left (\textit {\_a}^{2} b -a y \left (x \right )^{2}+b y \left (x \right )^{2}+a b -b^{2}\right )}}{a y \left (x \right )^{2}-\textit {\_a}^{2} b -b y \left (x \right )^{2}+\sqrt {a \left (\textit {\_a}^{2} b -a y \left (x \right )^{2}+b y \left (x \right )^{2}+a b -b^{2}\right )}\, \textit {\_a} -a b +b^{2}}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (-\frac {\textit {\_f} \left (a -b \right )}{a \,\textit {\_f}^{2}-b \,x^{2}-b \,\textit {\_f}^{2}+\sqrt {a \left (-a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+b \,x^{2}+a b -b^{2}\right )}\, x -a b +b^{2}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {\left (-\textit {\_a} b +\sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}\right ) \left (2 a \textit {\_f} -2 \textit {\_f} b +\frac {\textit {\_a} a \left (-2 a \textit {\_f} +2 \textit {\_f} b \right )}{2 \sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}}\right )}{\left (a \,\textit {\_f}^{2}-\textit {\_a}^{2} b -b \,\textit {\_f}^{2}+\sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}\, \textit {\_a} -a b +b^{2}\right )^{2}}-\frac {a \left (-2 a \textit {\_f} +2 \textit {\_f} b \right )}{2 \left (a \,\textit {\_f}^{2}-\textit {\_a}^{2} b -b \,\textit {\_f}^{2}+\sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}\, \textit {\_a} -a b +b^{2}\right ) \sqrt {a \left (\textit {\_a}^{2} b -a \,\textit {\_f}^{2}+b \,\textit {\_f}^{2}+a b -b^{2}\right )}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 1.39 (sec). Leaf size: 86
DSolve[(a-b) y[x]^2 (y'[x])^2 -2 b x y[x] y'[x]-a b -b x^2+a y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {b \left (b-x^2\right )+a \left (-b+(x-c_1){}^2\right )}}{\sqrt {b-a}} y(x)\to \frac {\sqrt {b \left (b-x^2\right )+a \left (-b+(x-c_1){}^2\right )}}{\sqrt {b-a}} \end{align*}