75.17.38 problem 588

Internal problem ID [17016]
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 : 588
Date solved : Monday, March 31, 2025 at 03:38:13 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}-y^{\prime \prime \prime \prime }&=x \,{\mathrm e}^{x}-1 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-diff(diff(diff(diff(y(x),x),x),x),x) = x*exp(x)-1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{2}+2 c_1 -8 x +20\right ) {\mathrm e}^{x}}{2}+\frac {x^{4}}{24}+\frac {c_2 \,x^{3}}{6}+\frac {c_3 \,x^{2}}{2}+c_4 x +c_5 \]

✓ Mathematica. Time used: 10.631 (sec). Leaf size: 148

ode=D[y[x],{x,5}]-D[y[x],{x,4}]==x*Exp[x]-1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \int _1^x\int _1^{K[4]}\int _1^{K[3]}\int _1^{K[2]}\left (e^{K[1]} \left (\frac {K[1]^2}{2}+c_1\right )+1\right )dK[1]dK[2]dK[3]dK[4]+x (x (c_5 x+c_4)+c_3)+c_2 \\ y(x)\to \frac {1}{24} \left (x^4+4 (-1+6 c_5) x^3+12 e^x \left (x^2-8 x+20\right )+6 (1+4 c_4) x^2-2 e \left (x^3+6 x^2+27 x+44\right )+4 (-1+6 c_3) x+1+24 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 37

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

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