Internal problem ID [2612]
Book: Differential equations with applications and historial notes, George F. Simmons,
1971
Section: Chapter 2, End of chapter, page 61
Problem number: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {\left ({\mathrm e}^{x}-3 x^{2} y^{2}\right ) y^{\prime }+{\mathrm e}^{x} y-2 x y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 347
dsolve((exp(x)-3*x^2*y(x)^2)*diff(y(x),x)+y(x)*exp(x)=2*x*y(x)^3,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}{6 x}+\frac {2 \,{\mathrm e}^{x}}{x \left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}{12 x}-\frac {{\mathrm e}^{x}}{x \left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}{6 x}-\frac {2 \,{\mathrm e}^{x}}{x \left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}{12 x}-\frac {{\mathrm e}^{x}}{x \left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}{6 x}-\frac {2 \,{\mathrm e}^{x}}{x \left (108 x c_{1}+12 \sqrt {81 x^{2} c_{1}^{2}-12 \,{\mathrm e}^{3 x}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 55.85 (sec). Leaf size: 364
DSolve[(Exp[x]-3*x^2*y[x]^2)*y'[x]+y[x]*Exp[x]==2*x*y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt [3]{3} e^x x^2+\sqrt [3]{2} \left (9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}\right ){}^{2/3}}{6^{2/3} x^2 \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}}{2 \sqrt [3]{2} 3^{2/3} x^2}-\frac {\left (\sqrt {3}+3 i\right ) e^x}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}}{2 \sqrt [3]{2} 3^{2/3} x^2}-\frac {\left (\sqrt {3}-3 i\right ) e^x}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\ \end{align*}