Internal problem ID [3876]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 628.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [`y=_G(x,y')`]
\[ \boxed {3 \left (x^{2}-y^{2}\right ) y^{\prime }+6 \left (x +1\right ) x y-2 y^{3}=-3 \,{\mathrm e}^{x}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 622
dsolve(3*(x^2-y(x)^2)*diff(y(x),x)+3*exp(x)+6*x*y(x)*(1+x)-2*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\mathrm e}^{-2 x} {\left (\left (4 \,{\mathrm e}^{3 x}+4 c_{1} +4 \sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {1}{3}}} y \left (x \right ) = -\frac {{\mathrm e}^{-2 x} {\left (\left (4 \,{\mathrm e}^{3 x}+4 c_{1} +4 \sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {{\mathrm e}^{-2 x} {\left (\left (4 \,{\mathrm e}^{3 x}+4 c_{1} +4 \sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +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 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_{1} +c_{1}^{2}}\right ) {\mathrm e}^{4 x}\right )}^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 60.281 (sec). Leaf size: 497
DSolve[3(x^2-y[x]^2)y'[x]+3 Exp[x]+6 x y[x](1+x)-2 y[x]^3==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*}