60.1.497 problem 500

Internal problem ID [10511]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 500
Date solved : Monday, January 27, 2025 at 08:31:46 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 y^{2}-b \,x^{2}-a b&=0 \end{align*}

✓ Solution by Maple

Time used: 0.313 (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*y(x)^2-b*x^2-a*b = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.425 (sec). Leaf size: 86

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