Internal problem ID [10109]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 8. Exact
differential equations. Page 11
Problem number: Ex 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\frac {y^{2}-2 x^{2}}{y^{2} x -x^{3}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 221
dsolve((y(x)^2-2*x^2)/(x*y(x)^2-x^3)+ (2*y(x)^2-x^2)/(y(x)^3-x^2*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {-x c_{1}-\frac {-2 x^{2} c_{1}^{2}+\sqrt {2 x^{4} c_{1}^{4}-2 \sqrt {x^{6} c_{1}^{6}+4}\, c_{1} x}}{2 x c_{1}}}{c_{1}} \\ y \relax (x ) = \frac {-x c_{1}-\frac {-2 x^{2} c_{1}^{2}+\sqrt {2 x^{4} c_{1}^{4}+2 \sqrt {x^{6} c_{1}^{6}+4}\, c_{1} x}}{2 x c_{1}}}{c_{1}} \\ y \relax (x ) = \frac {-x c_{1}+\frac {2 x^{2} c_{1}^{2}+\sqrt {2 x^{4} c_{1}^{4}-2 \sqrt {x^{6} c_{1}^{6}+4}\, c_{1} x}}{2 x c_{1}}}{c_{1}} \\ y \relax (x ) = \frac {-x c_{1}+\frac {2 x^{2} c_{1}^{2}+\sqrt {2 x^{4} c_{1}^{4}+2 \sqrt {x^{6} c_{1}^{6}+4}\, c_{1} x}}{2 x c_{1}}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 11.787 (sec). Leaf size: 277
DSolve[(y[x]^2-2*x^2)/(x*y[x]^2-x^3)+ (2*y[x]^2-x^2)/(y[x]^3-x^2*y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {x^3+\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {x^3+\sqrt {x^6-4 e^{2 c_1}}}{x}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2-\frac {\sqrt {x^6}}{x}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {\sqrt {x^6}+x^3}{x}}}{\sqrt {2}} \\ \end{align*}