60.5.36 problem 1573

Internal problem ID [11533]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1573
Date solved : Sunday, March 30, 2025 at 08:24:19 PM
CAS classification : [[_high_order, _fully, _exact, _linear]]

\begin{align*} \left ({\mathrm e}^{x}+2 x \right ) y^{\prime \prime \prime \prime }+4 \left ({\mathrm e}^{x}+2\right ) y^{\prime \prime \prime }+6 \,{\mathrm e}^{x} y^{\prime \prime }+4 \,{\mathrm e}^{x} y^{\prime }+y \,{\mathrm e}^{x}-\frac {1}{x^{5}}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=(exp(x)+2*x)*diff(diff(diff(diff(y(x),x),x),x),x)+4*(exp(x)+2)*diff(diff(diff(y(x),x),x),x)+6*exp(x)*diff(diff(y(x),x),x)+4*exp(x)*diff(y(x),x)+y(x)*exp(x)-1/x^5 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_4 +c_1 \,x^{3}+c_2 \,x^{2}+c_3 x +\frac {1}{24 x}}{{\mathrm e}^{x}+2 x} \]

✓ Mathematica. Time used: 0.086 (sec). Leaf size: 48

ode=-x^(-5) + E^x*y[x] + 4*E^x*D[y[x],x] + 6*E^x*D[y[x],{x,2}] + 4*(2 + E^x)*Derivative[3][y][x] + (E^x + 2*x)*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {24 c_4 x^4+24 c_3 x^3+24 c_2 x^2+24 c_1 x+1}{48 x^2+24 e^x x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x + exp(x))*Derivative(y(x), (x, 4)) + (4*exp(x) + 8)*Derivative(y(x), (x, 3)) + y(x)*exp(x) + 4*exp(x)*Derivative(y(x), x) + 6*exp(x)*Derivative(y(x), (x, 2)) - 1/x**5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*x**5*(x*Derivative(y(x), (x, 4)) + 4*Derivative(y(x), (x, 3))) - x**5*(y(x) + 6*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)))*exp(x) + 1)*exp(-x)/(4*x**5) cannot be solved by the factorable group method