29.24.14 problem 676

Internal problem ID [5267]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 676
Date solved : Monday, January 27, 2025 at 10:50:50 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 1.217 (sec). Leaf size: 381

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

\begin{align*} y \left (x \right ) &= \frac {x}{{\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {x}{{\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (\left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\ y \left (x \right ) &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) x^{3} c_{1} \right )}^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 6.764 (sec). Leaf size: 352

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

\begin{align*} y(x)\to \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to 0 \\ y(x)\to \sqrt [3]{x^3-\sqrt {x^6}} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6}} \\ y(x)\to \sqrt [3]{\sqrt {x^6}+x^3} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{\sqrt {x^6}+x^3} \\ y(x)\to (-1)^{2/3} \sqrt [3]{\sqrt {x^6}+x^3} \\ \end{align*}