82.9.1 problem Ex. 1

Internal problem ID [18753]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 26
Problem number : Ex. 1
Date solved : Tuesday, January 28, 2025 at 12:14:33 PM
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: 0.110 (sec). Leaf size: 71

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

\begin{align*} y \left (x \right ) &= \frac {\sqrt {\frac {2 c_{1} x^{3}-2 \sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ y \left (x \right ) &= \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: 12.028 (sec). Leaf size: 277

DSolve[(y[x]^3-2*y[x]*x^2)+(2*x*y[x]^2-x^3)*D[y[x],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*}