Internal problem ID [560]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.6. Page 100
Problem number: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class D], _rational]
Solve \begin {gather*} \boxed {2 y x +3 y x^{2}+y^{3}+\left (x^{2}+y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.034 (sec). Leaf size: 420
dsolve(2*x*y(x)+3*x^2*y(x)+y(x)^3+(x^2+y(x)^2)*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-3 x} \left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}{2 c_{1}}-\frac {2 x^{2} {\mathrm e}^{3 x} c_{1}}{\left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {{\mathrm e}^{-3 x} \left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}{4 c_{1}}+\frac {x^{2} {\mathrm e}^{3 x} c_{1}}{\left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\mathrm e}^{-3 x} \left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}{2 c_{1}}+\frac {2 x^{2} {\mathrm e}^{3 x} c_{1}}{\left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {{\mathrm e}^{-3 x} \left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}{4 c_{1}}+\frac {x^{2} {\mathrm e}^{3 x} c_{1}}{\left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\mathrm e}^{-3 x} \left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}{2 c_{1}}+\frac {2 x^{2} {\mathrm e}^{3 x} c_{1}}{\left (\left (4+4 \sqrt {4 x^{6} {\mathrm e}^{6 x} c_{1}^{2}+1}\right ) {\mathrm e}^{6 x} c_{1}^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.609 (sec). Leaf size: 544
DSolve[2*x*y[x]+3*x^2*y[x]+y[x]^3+(x^2+y[x]^2)*y'[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-3 x} \left (-2 e^{6 x} x^2+\sqrt [3]{2} \left (\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\ y(x)\to \frac {4 \sqrt [3]{-2} e^{3 x} x^2+2 (-2)^{2/3} e^{-3 x} \left (\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\ y(x)\to \frac {e^{-3 x} \left (\left (1-i \sqrt {3}\right ) e^{6 x} x^2-\sqrt [3]{-2} \left (\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\ y(x)\to \frac {e^{-3 x} \left (\sqrt [3]{e^{18 x} x^6}-e^{6 x} x^2\right )}{\sqrt [6]{e^{18 x} x^6}} \\ y(x)\to \frac {e^{-3 x} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{e^{18 x} x^6}+\left (1-i \sqrt {3}\right ) e^{6 x} x^2\right )}{2 \sqrt [6]{e^{18 x} x^6}} \\ y(x)\to \frac {e^{-3 x} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{e^{18 x} x^6}+\left (1+i \sqrt {3}\right ) e^{6 x} x^2\right )}{2 \sqrt [6]{e^{18 x} x^6}} \\ \end{align*}