69.16.27 problem 500

Internal problem ID [18287]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 500
Date solved : Thursday, October 02, 2025 at 03:10:08 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime }&={\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x) = exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{2}+\left (2 c_1 +4\right ) x +4 c_1 +2 c_2 +6\right ) {\mathrm e}^{-x}}{2}+c_3 x +c_4 \]

✓ Mathematica. Time used: 13.061 (sec). Leaf size: 158

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

\begin{align*} y(x)&\to \int _1^x\int _1^{K[2]}\frac {1}{2} e^{-K[1]} (2 c_1+K[1] (2 c_2+K[1]))dK[1]dK[2]+c_4 x+c_3\\ y(x)&\to e^{-x} \left (\frac {x^2}{2}+2 x+3+c_1\right )+\frac {\left (\frac {5}{2}+c_1\right ) x-2 (4+c_1)}{e}+c_4 x+c_3\\ y(x)&\to \frac {1}{2} e^{-x} \left (x^2+2 (2+c_2) x+6+4 c_2\right )+\frac {\left (\frac {5}{2}+2 c_2\right ) x-8-5 c_2}{e}+c_4 x+c_3 \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 22

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

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