75.16.28 problem 501

Internal problem ID [16930]
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 : 501
Date solved : Monday, March 31, 2025 at 03:35:44 PM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

\[ y = \frac {\left (24+x^{3}+6 x^{2}+6 \left (3+c_1 \right ) x +12 c_1 +6 c_2 \right ) {\mathrm e}^{-x}}{6}+c_3 x +c_4 \]

✓ Mathematica. Time used: 48.818 (sec). Leaf size: 168

ode=D[y[x],{x,4}]+2*D[y[x],{x,3}]+D[y[x],{x,2}]==x*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}{6} e^{-K[1]} \left (K[1]^3+6 c_2 K[1]+6 c_1\right )dK[1]dK[2]+c_4 x+c_3 \\ y(x)\to e^{-x} \left (\frac {x^3}{6}+x^2+3 x+4+c_1\right )+\frac {\left (\frac {8}{3}+c_1\right ) x-\frac {65}{6}-2 c_1}{e}+c_4 x+c_3 \\ y(x)\to e^{-x} \left (\frac {x^3}{6}+x^2+(3+c_2) x+2 (2+c_2)\right )+\frac {\left (\frac {8}{3}+2 c_2\right ) x-\frac {65}{6}-5 c_2}{e}+c_4 x+c_3 \\ \end{align*}

✓ Sympy. Time used: 0.191 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} + x \left (C_{2} + C_{3} e^{- x}\right ) + \left (C_{4} + x^{2} \left (\frac {x}{6} + 1\right )\right ) e^{- x} \]