Internal problem ID [3170]
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]
\[ \boxed {2 x \left (3 x +y-y \,{\mathrm e}^{-x^{2}}\right )+\left (x^{2}+3 y^{2}+{\mathrm e}^{-x^{2}}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 634
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 \left (x \right ) &= -\frac {12^{\frac {1}{3}} \left (-\left (\sqrt {3}\, {\mathrm e}^{2 x^{2}} \sqrt {\left (4+\left (112 x^{6}+108 c_{1} x^{3}+27 c_{1}^{2}\right ) {\mathrm e}^{3 x^{2}}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+12 \,{\mathrm e}^{x^{2}} x^{2}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{\frac {2}{3}} {\mathrm e}^{-x^{2}}+\left ({\mathrm e}^{x^{2}} x^{2}+1\right ) 12^{\frac {1}{3}}\right )}{6 \left (\sqrt {3}\, {\mathrm e}^{2 x^{2}} \sqrt {\left (4+\left (112 x^{6}+108 c_{1} x^{3}+27 c_{1}^{2}\right ) {\mathrm e}^{3 x^{2}}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+12 \,{\mathrm e}^{x^{2}} x^{2}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} \left ({\mathrm e}^{-x^{2}} \left (1+i \sqrt {3}\right ) \left (\sqrt {3}\, {\mathrm e}^{2 x^{2}} \sqrt {\left (4+\left (112 x^{6}+108 c_{1} x^{3}+27 c_{1}^{2}\right ) {\mathrm e}^{3 x^{2}}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+12 \,{\mathrm e}^{x^{2}} x^{2}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{\frac {2}{3}}+\left ({\mathrm e}^{x^{2}} x^{2}+1\right ) 2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right ) 2^{\frac {2}{3}}}{12 \left (\sqrt {3}\, {\mathrm e}^{2 x^{2}} \sqrt {\left (4+\left (112 x^{6}+108 c_{1} x^{3}+27 c_{1}^{2}\right ) {\mathrm e}^{3 x^{2}}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+12 \,{\mathrm e}^{x^{2}} x^{2}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\mathrm e}^{-x^{2}} \left (\sqrt {3}\, {\mathrm e}^{2 x^{2}} \sqrt {\left (4+\left (112 x^{6}+108 c_{1} x^{3}+27 c_{1}^{2}\right ) {\mathrm e}^{3 x^{2}}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+12 \,{\mathrm e}^{x^{2}} x^{2}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{\frac {2}{3}}+\left ({\mathrm e}^{x^{2}} x^{2}+1\right ) \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12 \left (\sqrt {3}\, {\mathrm e}^{2 x^{2}} \sqrt {\left (4+\left (112 x^{6}+108 c_{1} x^{3}+27 c_{1}^{2}\right ) {\mathrm e}^{3 x^{2}}+12 \,{\mathrm e}^{2 x^{2}} x^{4}+12 \,{\mathrm e}^{x^{2}} x^{2}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 37.566 (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*}