62.12.23 problem Ex 24

Internal problem ID [12869]
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 19. Summary. Page 29
Problem number : Ex 24
Date solved : Tuesday, January 28, 2025 at 04:29:14 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.629 (sec). Leaf size: 71

dsolve((y(x)^3-2*x^2*y(x))+(2*x*y(x)^2-x^3)*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: 0.142 (sec). Leaf size: 48

DSolve[(y[x]^3-2*x^2*y[x])+(2*x*y[x]^2-x^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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