Internal problem ID [1010]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable
Equations. Section 2.4 Page 68
Problem number: 33.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 129
dsolve(diff(y(x),x)=(x*y(x)^2+2*y(x)^3)/(x^3+x^2*y(x)+x*y(x)^2),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \RootOf \left (\textit {\_Z}^{8} x^{2} c_{1}+2 \textit {\_Z}^{6} x^{2} c_{1}+\textit {\_Z}^{4} x^{2} c_{1}-2 \textit {\_Z}^{2}-1\right )^{6} x^{3}+2 c_{1} \RootOf \left (\textit {\_Z}^{8} x^{2} c_{1}+2 \textit {\_Z}^{6} x^{2} c_{1}+\textit {\_Z}^{4} x^{2} c_{1}-2 \textit {\_Z}^{2}-1\right )^{4} x^{3}+c_{1} \RootOf \left (\textit {\_Z}^{8} x^{2} c_{1}+2 \textit {\_Z}^{6} x^{2} c_{1}+\textit {\_Z}^{4} x^{2} c_{1}-2 \textit {\_Z}^{2}-1\right )^{2} x^{3}-x \]
✓ Solution by Mathematica
Time used: 60.171 (sec). Leaf size: 1989
DSolve[y'[x]==(x*y[x]^2+2*y[x]^3)/(x^3+x^2*y[x]+x*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}+\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\ y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}+\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}-\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt {3} \sqrt {-4 e^{2 c_1} x^4+6 x^2-\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}-\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}-\frac {6 \sqrt {3} x^3 \left (1+e^{2 c_1} x^2\right )}{\sqrt {-2 e^{2 c_1} x^4+3 x^2+\frac {e^{4 c_1} x^8}{\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}+\sqrt [3]{e^{6 c_1} x^{12}+54 e^{2 c_1} x^8+6 \sqrt {3} \sqrt {e^{4 c_1} x^{16} \left (27+e^{4 c_1} x^4\right )}}}}}+3 x\right ) \\ \end{align*}