48.3.7 problem Example 3.36

Internal problem ID [7531]
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 3.36
Date solved : Sunday, March 30, 2025 at 12:13:49 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y&=g \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 64

ode:=diff(diff(diff(y(t),t),t),t)-diff(diff(y(t),t),t)-diff(y(t),t)+y(t) = g(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {\int \left (2 t +1\right ) g \left (t \right ) {\mathrm e}^{-t}d t {\mathrm e}^{t}}{4}+\frac {\int {\mathrm e}^{-t} g \left (t \right )d t {\mathrm e}^{t} t}{2}+\frac {\int {\mathrm e}^{t} g \left (t \right )d t {\mathrm e}^{-t}}{4}+c_2 \,{\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_3 t +c_1 \right ) \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 106

ode=D[ y[t],{t,3}]-D[y[t],{t,2}]-D[y[t],t]+y[t]==g[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-t} \int _1^t\frac {1}{4} e^{K[1]} g(K[1])dK[1]+e^t t \int _1^t\frac {1}{2} e^{-K[3]} g(K[3])dK[3]+e^t \int _1^t-\frac {1}{4} e^{-K[2]} g(K[2]) (2 K[2]+1)dK[2]+c_1 e^{-t}+c_2 e^t+c_3 e^t t \]

✓ Sympy. Time used: 5.270 (sec). Leaf size: 51

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

\[ y{\left (t \right )} = \left (C_{1} + \frac {\int g{\left (t \right )} e^{t}\, dt}{4}\right ) e^{- t} + \left (C_{2} + t \left (C_{3} + \frac {\int g{\left (t \right )} e^{- t}\, dt}{2}\right ) - \frac {\int \left (2 t + 1\right ) g{\left (t \right )} e^{- t}\, dt}{4}\right ) e^{t} \]