29.34.3 problem 998

Internal problem ID [5574]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 998
Date solved : Monday, January 27, 2025 at 12:10:00 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} \left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }-a b -b \,x^{2}+a y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.342 (sec). Leaf size: 766

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 {-b \textit {\_a} -\sqrt {a \left (\left (b -a \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}}{\sqrt {a \left (\left (b -a \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}\, \textit {\_a} +\left (b -a \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {\textit {\_f} \left (\left (\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (b -a \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (2 b \textit {\_a} \sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}+a \left (-b^{2}+\left (2 \textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )\right ) \left (a -b \right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, {\left (\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, \textit {\_a} -b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}^{2}}d \textit {\_a} \right )+a -b \right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (b -a \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )}d \textit {\_f} +c_{1} &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {b \textit {\_a} -\sqrt {a \left (\left (b -a \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}}{-\sqrt {a \left (\left (b -a \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}\, \textit {\_a} +\left (b -a \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {\left (\left (-\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (b -a \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (-2 b \textit {\_a} \sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}+a \left (-b^{2}+\left (2 \textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )\right ) \left (a -b \right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, {\left (-\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, \textit {\_a} -b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}^{2}}d \textit {\_a} \right )+a -b \right ) \textit {\_f}}{-\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (b -a \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )}d \textit {\_f} +c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.426 (sec). Leaf size: 86

DSolve[(a-b) y[x]^2 (D[y[x],x])^2 -2 b x y[x] D[y[x],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*}