60.2.220 problem 796
Internal
problem
ID
[10807]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
796
Date
solved
:
Tuesday, January 28, 2025 at 05:16:33 PM
CAS
classification
:
[[_Abel, `2nd type`, `class C`]]
\begin{align*} y^{\prime }&=\frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+9 y} \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 135
dsolve(diff(y(x),x) = 1/3*y(x)^3*x*exp(3*x^2)/(3*exp(3/2*x^2)+exp(3/2*x^2)*y(x)+3*y(x))/exp(9/2*x^2),y(x), singsol=all)
\[
y = \operatorname {RootOf}\left (81+\left (27 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+7 \,{\mathrm e}^{3 x^{2}+\operatorname {RootOf}\left (\left (42 \sqrt {93}\, \sinh \left (\frac {\left (c_{1} -5 \textit {\_Z} \right ) \sqrt {93}}{90}\right ) {\mathrm e}^{3 x^{2}+\textit {\_Z}} \cosh \left (\frac {\left (c_{1} -5 \textit {\_Z} \right ) \sqrt {93}}{90}\right )+406 \,{\mathrm e}^{3 x^{2}+\textit {\_Z}} \cosh \left (\frac {\left (c_{1} -5 \textit {\_Z} \right ) \sqrt {93}}{90}\right )^{2}-217 \,{\mathrm e}^{3 x^{2}+\textit {\_Z}}+93\right ) {\mathrm e}^{3 x^{2}}\right )}+9 \,{\mathrm e}^{3 x^{2}}-3\right ) \textit {\_Z}^{2}+\left (54 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81\right ) \textit {\_Z} \right ) {\mathrm e}^{\frac {3 x^{2}}{2}}
\]
✓ Solution by Mathematica
Time used: 7.053 (sec). Leaf size: 218
DSolve[D[y[x],x] == (x*y[x]^3)/(3*E^((3*x^2)/2)*(3*E^((3*x^2)/2) + 3*y[x] + E^((3*x^2)/2)*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{-\frac {e^{3 x^2} x \left (\left (10+3 e^{\frac {3 x^2}{2}}\right ) y(x)+9 e^{\frac {3 x^2}{2}}\right )}{3^{2/3} \sqrt [3]{7} \left (3+e^{\frac {3 x^2}{2}}\right ) \sqrt [3]{-\frac {e^{9 x^2} x^3}{\left (3+e^{\frac {3 x^2}{2}}\right )^3}} \left (\left (3+e^{\frac {3 x^2}{2}}\right ) y(x)+3 e^{\frac {3 x^2}{2}}\right )}}\frac {1}{K[1]^3+\frac {10 \sqrt [3]{-\frac {1}{3}} K[1]}{7^{2/3}}+1}dK[1]=\frac {3}{2} \sqrt [3]{3} 7^{2/3} e^{-6 x^2} \left (-\frac {e^{9 x^2} x^3}{\left (e^{\frac {3 x^2}{2}}+3\right )^3}\right )^{2/3} \left (e^{\frac {3 x^2}{2}}+3\right )^2+c_1,y(x)\right ]
\]