29.24.23 problem 685

Internal problem ID [5276]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 24
Problem number : 685
Date solved : Monday, January 27, 2025 at 10:56:04 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.944 (sec). Leaf size: 139

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

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-8 c_{1}^{2} x^{2}-2 \sqrt {8 c_{1}^{2} x^{2}+1}+2}}{4 c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {-8 c_{1}^{2} x^{2}-2 \sqrt {8 c_{1}^{2} x^{2}+1}+2}}{4 c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-4 c_{1}^{2} x^{2}+1+\sqrt {8 c_{1}^{2} x^{2}+1}}}{4 c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-4 c_{1}^{2} x^{2}+1+\sqrt {8 c_{1}^{2} x^{2}+1}}}{4 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 22.171 (sec). Leaf size: 253

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

\begin{align*} y(x)\to -\frac {\sqrt {-4 x^2-\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {-4 x^2-\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {-4 x^2+\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {-4 x^2+\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}} \\ y(x)\to \text {Undefined} \\ y(x)\to -\frac {\sqrt {-x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-x^2}}{\sqrt {2}} \\ \end{align*}