75.16.25 problem 498

Internal problem ID [16927]
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 : 498
Date solved : Monday, March 31, 2025 at 03:35:41 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+4 y^{\prime \prime }&=1 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (2 c_1 x +2 c_1 +2 c_2 \right ) {\mathrm e}^{-2 x}}{8}+\frac {x^{2}}{8}+c_3 x +c_4 \]

✓ Mathematica. Time used: 28.187 (sec). Leaf size: 133

ode=D[y[x],{x,4}]+4*D[y[x],{x,3}]+4*D[y[x],{x,2}]==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.118 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{4} e^{- 2 x} + \frac {x^{2}}{8} + x \left (C_{2} + C_{3} e^{- 2 x}\right ) \]