Internal problem ID [15076]
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: 188.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {3 x^{2} y+y^{3}+\left (x^{3}+3 y^{2} x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 274
dsolve(( 3*x^2*y(x)+y(x)^3)+(x^3+3*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {12^{\frac {1}{3}} \left (x^{4} c_{1}^{2} 12^{\frac {1}{3}}-{\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {2}{3}}\right )}{6 c_{1} x {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {2}{3}}+c_{1}^{2} 2^{\frac {2}{3}} x^{4} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {1}{3}} x c_{1}} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {2}{3}}+c_{1}^{2} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{4}\right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {1}{3}} x c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.214 (sec). Leaf size: 338
DSolve[( 3*x^2*y[x]+y[x]^3)+(x^3+3*x*y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}}} \\ y(x)\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^2}{12 \sqrt [3]{\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}}} \\ y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^2-i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}\right ){}^{2/3}}{12 \sqrt [3]{\frac {\sqrt {12 x^8+81 e^{2 c_1}}+9 e^{c_1}}{x}}} \\ \end{align*}