69.17.39 problem 589

Internal problem ID [18375]
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. Superposition principle. Exercises page 137
Problem number : 589
Date solved : Thursday, October 02, 2025 at 03:11:06 PM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

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

\[ y = \frac {\left (7+2 x -2 c_1 \right ) {\mathrm e}^{-x}}{2}-\frac {x^{4}}{24}+\frac {c_3 \,x^{2}}{2}+c_4 x +{\mathrm e}^{x} c_2 +c_5 \]

✓ Mathematica. Time used: 40.287 (sec). Leaf size: 113

ode=D[y[x],{x,5}]-D[y[x],{x,3}]==x+2*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\int _1^{K[5]}\int _1^{K[4]}e^{-K[3]} \left (e^{2 K[3]} c_1+c_2+e^{2 K[3]} \int _1^{K[3]}\frac {1}{2} e^{-2 K[1]} \left (e^{K[1]} K[1]+2\right )dK[1]+\int _1^{K[3]}\left (-\frac {1}{2} e^{K[2]} K[2]-1\right )dK[2]\right )dK[3]dK[4]dK[5]+x (c_5 x+c_4)+c_3 \end{align*}

✓ Sympy. Time used: 0.081 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} + C_{2} x^{2} + C_{4} e^{- x} + C_{5} e^{x} - \frac {x^{4}}{24} + x \left (C_{3} + e^{- x}\right ) \]