29.23.19 problem 650

Internal problem ID [5241]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 650
Date solved : Monday, January 27, 2025 at 10:43:05 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.953 (sec). Leaf size: 89

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

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

✓ Solution by Mathematica

Time used: 43.871 (sec). Leaf size: 277

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

\begin{align*} y(x)\to -\frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-x^2+\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}} \\ y(x)\to \sqrt {-\frac {x^2}{2}+\frac {1}{2} \sqrt {x^4+4 e^{2 c_1} x^6}} \\ y(x)\to -\frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}} \\ \end{align*}