42.3.10 problem Example 3.39

Internal problem ID [8820]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page 181
Problem number : Example 3.39
Date solved : Tuesday, September 30, 2025 at 05:52:55 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-4 y&=f \left (x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 81

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)-4*diff(y(x),x)-4*y(x) = f(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-2 x} \left (\left (\int {\mathrm e}^{-x} f \left (x \right )d x +18 c_1 \right ) {\mathrm e}^{3 x}+9 \left (-\int {\mathrm e}^{x} f \left (x \right )d x +2 c_3 \right ) {\mathrm e}^{x}+6 \int f \left (x \right ) {\mathrm e}^{2 x}d x x +18 c_4 x -2 \int \left (3 x -4\right ) f \left (x \right ) {\mathrm e}^{2 x}d x +18 c_2 \right )}{18} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 128

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

\begin{align*} y(x)&\to e^{-2 x} \left (x \int _1^x\frac {1}{3} e^{2 K[2]} f(K[2])dK[2]+e^x \int _1^x-\frac {1}{2} e^{K[3]} f(K[3])dK[3]+e^{3 x} \int _1^x\frac {1}{18} e^{-K[4]} f(K[4])dK[4]+\int _1^x-\frac {1}{9} e^{2 K[1]} f(K[1]) (3 K[1]-4)dK[1]+c_2 x+c_3 e^x+c_4 e^{3 x}+c_1\right ) \end{align*}

✓ Sympy. Time used: 1.981 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
f = Function("f") 
ode = Eq(-f(x) - 4*y(x) - 4*Derivative(y(x), x) + 3*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)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {\int f{\left (x \right )} e^{x}\, dx}{2}\right ) e^{- x} + \left (C_{2} + \frac {\int f{\left (x \right )} e^{- x}\, dx}{18}\right ) e^{x} + \left (C_{3} + x \left (C_{4} + \frac {\int f{\left (x \right )} e^{2 x}\, dx}{3}\right ) - \frac {\int \left (3 x - 4\right ) f{\left (x \right )} e^{2 x}\, dx}{9}\right ) e^{- 2 x} \]