62.1.2 problem Ex 2

Internal problem ID [12799]
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
Date solved : Tuesday, January 28, 2025 at 04:20:19 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \frac {y^{2}-2 x^{2}}{x y^{2}-x^{3}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.651 (sec). Leaf size: 71

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 &= \frac {\sqrt {\frac {2 c_{1} x^{3}-2 \sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ y &= \frac {\sqrt {2}\, \sqrt {\frac {c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 48

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])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {2 K[1]^2-1}{(K[1]-1) K[1] (K[1]+1)}dK[1]=-3 \log (x)+c_1,y(x)\right ] \]