Internal problem ID [7863]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 283.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {3 \left (y^{2}-x^{2}\right ) y^{\prime }+2 y^{3}-6 x \left (1+x \right ) y-3 \,{\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 622
dsolve(3*(y(x)^2-x^2)*diff(y(x),x)+2*y(x)^3-6*x*(x+1)*y(x)-3*exp(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-2 x} \left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2} {\mathrm e}^{2 x}}{\left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {{\mathrm e}^{-2 x} \left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}{4}-\frac {x^{2} {\mathrm e}^{2 x}}{\left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\mathrm e}^{-2 x} \left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} {\mathrm e}^{2 x}}{\left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {{\mathrm e}^{-2 x} \left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}{4}-\frac {x^{2} {\mathrm e}^{2 x}}{\left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\mathrm e}^{-2 x} \left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2} {\mathrm e}^{2 x}}{\left (\left (4 \,{\mathrm e}^{3 x}-4 c_{1}+4 \sqrt {-4 \,{\mathrm e}^{4 x} x^{6}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.627 (sec). Leaf size: 479
DSolve[3*(y[x]^2-x^2)*y'[x]+2*y[x]^3-6*x*(x+1)*y[x]-3*Exp[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} e^{2 x} x^2}{\sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\ y(x)\to \frac {e^{-2 x} \left (\left (\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}\right ){}^{2/3} \text {Root}\left [\text {$\#$1}^3+16\&,2\right ]+\left (2+2 i \sqrt {3}\right ) e^{4 x} x^2\right )}{2\ 2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\ y(x)\to \frac {e^{-2 x} \left (\sqrt [3]{-2} \left (\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}\right ){}^{2/3}+\left (1-i \sqrt {3}\right ) e^{4 x} x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\ \end{align*}