62.3.2 problem Ex 2

Internal problem ID [12808]
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 10. Homogeneous equations. Page 15
Problem number : Ex 2
Date solved : Tuesday, January 28, 2025 at 04:22:19 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 1.848 (sec). Leaf size: 89

dsolve((2*x^2*y(x)+3*y(x)^3)-(x^3+2*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.124 (sec). Leaf size: 42

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

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