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: 0.015 (sec). Leaf size: 345
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 {2^{\frac {1}{3}} \left (2 \,{\mathrm e}^{4 x} x^{2}+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 \,{\mathrm e}^{4 x} \left (i \sqrt {3}-1\right ) x^{2}+\left (1+i \sqrt {3}\right ) 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^{\frac {1}{3}}}{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 {{\mathrm e}^{-2 x} 2^{\frac {1}{3}} \left (-2 \,{\mathrm e}^{4 x} \left (1+i \sqrt {3}\right ) x^{2}+\left (i \sqrt {3}-1\right ) 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 )}{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: 60.281 (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*}