54.2.238 problem 815

Internal problem ID [12112]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 815
Date solved : Sunday, October 12, 2025 at 02:19:48 AM
CAS classification : [[_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 157

ode:=diff(y(x),x) = 1/81*(3+y(x))^3*exp(9/2*x^2)*x*exp(3/2*x^2)/(3*exp(3/2*x^2)+exp(3/2*x^2)*y(x)+3*y(x))/exp(3*x^2); 
dsolve(ode,y(x), singsol=all);
 

\[ 5 \ln \left (3\right )-10 \ln \left (\frac {3+y}{3 \,{\mathrm e}^{-\frac {3 x^{2}}{2}} y+3+y}\right )-5 \ln \left (7\right )+5 \ln \left (\frac {\left (-81 y^{2}-243 y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}+\left (3+y\right )^{2} {\mathrm e}^{3 x^{2}}-243 y^{2}}{\left ({\mathrm e}^{\frac {3 x^{2}}{2}} \left (3+y\right )+3 y\right )^{2}}\right )-\frac {30 \sqrt {93}\, \operatorname {arctanh}\left (\frac {\left (29 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+87 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 y\right ) \sqrt {93}}{\left (279 y+837\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}+837 y}\right )}{31}+15 x^{2}-c_1 = 0 \]

✓ Mathematica. Time used: 5.95 (sec). Leaf size: 226

ode=D[y[x],x] == (E^(3*x^2)*x*(3 + y[x])^3)/(81*(3*E^((3*x^2)/2) + 3*y[x] + E^((3*x^2)/2)*y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {-30 e^{3 x^2} x-e^{\frac {3 x^2}{2}} \left (27+10 e^{\frac {3 x^2}{2}}\right ) y(x) x}{3\ 3^{2/3} \sqrt [3]{7} \left (3+e^{\frac {3 x^2}{2}}\right ) \sqrt [3]{-\frac {e^{\frac {9 x^2}{2}} x^3}{\left (3+e^{\frac {3 x^2}{2}}\right )^3}} \left (\left (3+e^{\frac {3 x^2}{2}}\right ) y(x)+3 e^{\frac {3 x^2}{2}}\right )}}\frac {1}{K[1]^3+\frac {10 \sqrt [3]{-\frac {1}{3}} K[1]}{7^{2/3}}+1}dK[1]+\frac {3}{2} \sqrt [3]{3} 7^{2/3} e^{-3 x^2} \left (-\frac {e^{\frac {9 x^2}{2}} x^3}{\left (e^{\frac {3 x^2}{2}}+3\right )^3}\right )^{2/3} \left (e^{\frac {3 x^2}{2}}+3\right )^2=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(y(x) + 3)**3*exp(3*x**2)/(81*y(x)*exp(3*x**2/2) + 243*y(x) + 243*exp(3*x**2/2)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out