Internal problem ID [2661]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {2 x \left (3 x +y-y \,{\mathrm e}^{-x^{2}}\right )+\left (x^{2}+3 y^{2}+{\mathrm e}^{-x^{2}}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1085
dsolve((2*x*(3*x+y(x)-y(x)*exp(-x^2)))+(x^2+3*y(x)^2+exp(-x^2))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-x^{2}} \left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 \left (x^{2} {\mathrm e}^{x^{2}}+1\right )}{\left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {{\mathrm e}^{-x^{2}} \left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{2} {\mathrm e}^{x^{2}}+1}{\left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\mathrm e}^{-x^{2}} \left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} {\mathrm e}^{x^{2}}+2}{\left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {{\mathrm e}^{-x^{2}} \left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}{12}+\frac {x^{2} {\mathrm e}^{x^{2}}+1}{\left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\mathrm e}^{-x^{2}} \left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2} {\mathrm e}^{x^{2}}+2}{\left (\left (-216 x^{3} {\mathrm e}^{x^{2}}-108 c_{1} {\mathrm e}^{x^{2}}+12 \sqrt {3}\, \sqrt {\left (112 \,{\mathrm e}^{3 x^{2}} x^{6}+108 \,{\mathrm e}^{3 x^{2}} c_{1} x^{3}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+27 \,{\mathrm e}^{3 x^{2}} c_{1}^{2}+12 x^{2} {\mathrm e}^{x^{2}}+4\right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{2 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 35.548 (sec). Leaf size: 416
DSolve[(2*x*(3*x+y[x]-y[x]*Exp[-x^2]))+(x^2+3*y[x]^2+Exp[-x^2])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-6 \sqrt [3]{2} \left (x^2+e^{-x^2}\right )+2^{2/3} \left (-54 x^3+\sqrt {108 \left (x^2+e^{-x^2}\right )^3+729 \left (-2 x^3+c_1\right ){}^2}+27 c_1\right ){}^{2/3}}{6 \sqrt [3]{-54 x^3+\sqrt {108 \left (x^2+e^{-x^2}\right )^3+729 \left (-2 x^3+c_1\right ){}^2}+27 c_1}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^2+e^{-x^2}\right )}{2^{2/3} \sqrt [3]{-54 x^3+\sqrt {108 \left (x^2+e^{-x^2}\right )^3+729 \left (-2 x^3+c_1\right ){}^2}+27 c_1}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+\sqrt {108 \left (x^2+e^{-x^2}\right )^3+729 \left (-2 x^3+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^2+e^{-x^2}\right )}{2^{2/3} \sqrt [3]{-54 x^3+\sqrt {108 \left (x^2+e^{-x^2}\right )^3+729 \left (-2 x^3+c_1\right ){}^2}+27 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+\sqrt {108 \left (x^2+e^{-x^2}\right )^3+729 \left (-2 x^3+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \\ \end{align*}