Internal problem ID [8619]
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')`]
\[ \boxed {3 \left (y^{2}-x^{2}\right ) y^{\prime }+2 y^{3}-6 \left (1+x \right ) y x=3 \,{\mathrm e}^{x}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 357
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 \left (x \right ) &= \frac {2^{\frac {1}{3}} \left (2 x^{2} {\mathrm e}^{4 x}+2^{\frac {1}{3}} {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {2}{3}}\right ) {\mathrm e}^{-2 x}}{2 {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (-2 x^{2} {\mathrm e}^{4 x} \left (i \sqrt {3}-1\right )+2^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} {\mathrm e}^{-2 x}}{4 {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} {\mathrm e}^{-2 x} \left (-2 x^{2} {\mathrm e}^{4 x} \left (1+i \sqrt {3}\right )+2^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {2}{3}}\right )}{4 {\left (\left ({\mathrm e}^{3 x}-c_{1} +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}-2 c_{1} {\mathrm e}^{3 x}+c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.3 (sec). Leaf size: 497
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 {\left (1-i \sqrt {3}\right ) 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}}}{2 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) e^{2 x} x^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 {\left (1+i \sqrt {3}\right ) 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}}}{2 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) e^{2 x} x^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}}} \\ \end{align*}