89.19.22 problem 22

Internal problem ID [24750]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 151
Problem number : 22
Date solved : Thursday, October 02, 2025 at 10:47:40 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+4 y^{\prime \prime }&=2 \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

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) = 2*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to \frac {1}{4} e^{2 x} \left (x^2+(-2+c_2) x+\frac {3}{2}+c_1-c_2\right )+c_4 x+c_3 \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 24

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

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