60.2.99 problem 675

Internal problem ID [10673]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 675
Date solved : Sunday, March 30, 2025 at 06:19:27 PM
CAS classification : [[_homogeneous, `class D`], _Riccati]

\begin{align*} y^{\prime }&=\frac {y+x^{3} a \,{\mathrm e}^{x}+a \,x^{4}+x^{3} a -x y^{2} {\mathrm e}^{x}-x^{2} y^{2}-x y^{2}}{x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

ode:=diff(y(x),x) = (y(x)+x^3*a*exp(x)+a*x^4+a*x^3-x*y(x)^2*exp(x)-x^2*y(x)^2-x*y(x)^2)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \tanh \left (\frac {\left (\left (6 x -6\right ) {\mathrm e}^{x}+2 x^{3}+3 x^{2}+6 c_1 \right ) \sqrt {a}}{6}\right ) x \sqrt {a} \]

✓ Mathematica. Time used: 0.292 (sec). Leaf size: 47

ode=D[y[x],x] == (a*x^3 + a*E^x*x^3 + a*x^4 + y[x] - x*y[x]^2 - E^x*x*y[x]^2 - x^2*y[x]^2)/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{a-K[1]^2}dK[1]=\int _1^xK[2] \left (K[2]+e^{K[2]}+1\right )dK[2]+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (a*x**4 + a*x**3*exp(x) + a*x**3 - x**2*y(x)**2 - x*y(x)**2*exp(x) - x*y(x)**2 + y(x))/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*x**3 - a*x**2*exp(x) - a*x**2 + x*y(x)**2 + y(x)**2*exp(x) + y(x)**2 + Derivative(y(x), x) - y(x)/x cannot be solved by the factorable group method