75.12.4 problem 278

Internal problem ID [16870]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 278
Date solved : Tuesday, January 28, 2025 at 09:36:13 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.361 (sec). Leaf size: 119

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

\begin{align*} y &= \frac {\sqrt {3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y &= \frac {\sqrt {3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y &= -\frac {\sqrt {3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y &= -\frac {\sqrt {3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.925 (sec). Leaf size: 245

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

\begin{align*} y(x)\to -\sqrt {3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {3 x^2-2 \sqrt {2} \sqrt {x^4}} \\ y(x)\to \sqrt {3 x^2-2 \sqrt {2} \sqrt {x^4}} \\ y(x)\to -\sqrt {2 \sqrt {2} \sqrt {x^4}+3 x^2} \\ y(x)\to \sqrt {2 \sqrt {2} \sqrt {x^4}+3 x^2} \\ \end{align*}