28.1.25 problem 25
Internal
problem
ID
[4331]
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
Date
solved
:
Monday, January 27, 2025 at 09:03:28 AM
CAS
classification
:
[_exact]
\begin{align*} 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{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 630
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^{{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 x^{2} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{{2}/{3}} {\mathrm e}^{-x^{2}}+\left (x^{2} {\mathrm e}^{x^{2}}+1\right ) 12^{{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 x^{2} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} 3^{{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 x^{2} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{{2}/{3}}+\left (x^{2} {\mathrm e}^{x^{2}}+1\right ) 2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{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 x^{2} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left ({\mathrm e}^{-x^{2}} \left (i \sqrt {3}-1\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 x^{2} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{{2}/{3}}+\left (x^{2} {\mathrm e}^{x^{2}}+1\right ) 2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right )}{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 x^{2} {\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-x^{2}}}-18 \,{\mathrm e}^{3 x^{2}} \left (x^{3}+\frac {c_{1}}{2}\right )\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 27.567 (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])*D[y[x],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*}