29.21.3 problem 579

Internal problem ID [5173]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 579
Date solved : Monday, January 27, 2025 at 10:17:15 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

\begin{align*} x^{2} \left (x -2 y\right ) y^{\prime }&=2 x^{3}-4 x y^{2}+y^{3} \end{align*}

✓ Solution by Maple

Time used: 0.894 (sec). Leaf size: 65

dsolve(x^2*(x-2*y(x))*diff(y(x),x) = 2*x^3-4*x*y(x)^2+y(x)^3,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {x \left (2 c_{1} x^{2}-\sqrt {3 c_{1} x^{2}+1}\right )}{c_{1} x^{2}-1} \\ y \left (x \right ) &= \frac {x \left (2 c_{1} x^{2}+\sqrt {3 c_{1} x^{2}+1}\right )}{c_{1} x^{2}-1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 14.095 (sec). Leaf size: 132

DSolve[x^2(x-2 y[x])D[y[x],x]==2 x^3-4 x y[x]^2+y[x]^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2 x^3-\sqrt {e^{2 c_1} x^2 \left (-3 x^2+e^{2 c_1}\right )}}{x^2+e^{2 c_1}} \\ y(x)\to \frac {2 x^3+\sqrt {e^{2 c_1} x^2 \left (-3 x^2+e^{2 c_1}\right )}}{x^2+e^{2 c_1}} \\ y(x)\to 2 x \\ y(x)\to -\sqrt {x^2} \\ y(x)\to \sqrt {x^2} \\ \end{align*}