Internal problem ID [3354]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 612.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {\left (x^{3}+2 y-y^{2}\right ) y^{\prime }+3 x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 493
dsolve((x^3+2*y(x)-y(x)^2)*diff(y(x),x)+3*x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{2}-\frac {2 \left (-x^{3}-1\right )}{\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}+1 \\ y \relax (x ) = -\frac {\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {-2 x^{3}-2}{\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {-2 x^{3}-2}{\left (12 x^{3}+12 c_{1}+8+4 \sqrt {-4 x^{9}-3 x^{6}+18 x^{3} c_{1}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.242 (sec). Leaf size: 409
DSolve[(x^3+2 y[x]-y[x]^2)y'[x]+3 x^2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{2} \left (x^3+1\right )}{\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}-\frac {\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{\sqrt [3]{2}}+1 \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1 \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1 \\ y(x)\to 0 \\ \end{align*}