83.12.5 problem 5

Internal problem ID [22006]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter VII. Operational method. Ex. XV at page 121
Problem number : 5
Date solved : Thursday, October 02, 2025 at 08:21:37 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }+y^{\prime \prime }-y^{\prime }&=x \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 30

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x)-diff(y(x),x) = x*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{2}+4 c_2 -4 x +4\right ) {\mathrm e}^{x}}{4}+c_1 \sin \left (x \right )-c_3 \cos \left (x \right )+c_4 \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 38

ode=D[y[x],{x,4}]-D[y[x],{x,3}]+D[y[x],{x,2}]-D[y[x],x]==x*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^x \left (x^2-4 x+5+4 c_3\right )-c_2 \cos (x)+c_1 \sin (x)+c_4 \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} + C_{3} \sin {\left (x \right )} + C_{4} \cos {\left (x \right )} + \left (C_{2} + \frac {x^{2}}{4} - x\right ) e^{x} \]