75.7.1 problem 175

Internal problem ID [16794]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 175
Date solved : Tuesday, January 28, 2025 at 09:23:16 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.339 (sec). Leaf size: 125

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

\begin{align*} y &= -\frac {\sqrt {-2 c_{1} x^{2}-2 \sqrt {-3 c_{1}^{2} x^{4}+4}}}{2 \sqrt {c_{1}}} \\ y &= \frac {\sqrt {-2 c_{1} x^{2}-2 \sqrt {-3 c_{1}^{2} x^{4}+4}}}{2 \sqrt {c_{1}}} \\ y &= -\frac {\sqrt {-2 c_{1} x^{2}+2 \sqrt {-3 c_{1}^{2} x^{4}+4}}}{2 \sqrt {c_{1}}} \\ y &= \frac {\sqrt {-2 c_{1} x^{2}+2 \sqrt {-3 c_{1}^{2} x^{4}+4}}}{2 \sqrt {c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 25.124 (sec). Leaf size: 303

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

\begin{align*} y(x)\to -\frac {\sqrt {-x^2-\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-x^2-\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-x^2+\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-x^2+\sqrt {-3 x^4+4 e^{2 c_1}}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {-\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {\sqrt {3} \sqrt {-x^4}-x^2}}{\sqrt {2}} \\ \end{align*}