67.4.39 problem Problem 13(c)

Internal problem ID [14012]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 13(c)
Date solved : Monday, March 31, 2025 at 08:21:49 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y&=8 \,{\mathrm e}^{2 t}-5 \,{\mathrm e}^{t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.109 (sec). Leaf size: 22

ode:=diff(diff(diff(y(t),t),t),t)-diff(diff(y(t),t),t)+4*diff(y(t),t)-4*y(t) = 8*exp(2*t)-5*exp(t); 
ic:=y(0) = 2, D(y)(0) = 0, (D@@2)(y)(0) = 3; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\sin \left (2 t \right )+{\mathrm e}^{2 t}-{\mathrm e}^{t} \left (-1+t \right ) \]

✓ Mathematica. Time used: 0.426 (sec). Leaf size: 212

ode=D[ y[t],{t,3}]-D[y[t],{t,2}]+4*D[y[t],t]-4*y[t]==8*Exp[2*t]-5*Exp[t]; 
ic={y[0]==2,Derivative[1][y][0] ==0,Derivative[2][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{5} \left (-5 \cos (2 t) \int _1^0-\frac {1}{10} e^{K[1]} \left (-5+8 e^{K[1]}\right ) (2 \cos (2 K[1])-\sin (2 K[1]))dK[1]+5 \cos (2 t) \int _1^t-\frac {1}{10} e^{K[1]} \left (-5+8 e^{K[1]}\right ) (2 \cos (2 K[1])-\sin (2 K[1]))dK[1]-5 \sin (2 t) \int _1^0-\frac {1}{10} e^{K[2]} \left (-5+8 e^{K[2]}\right ) (\cos (2 K[2])+2 \sin (2 K[2]))dK[2]+5 \sin (2 t) \int _1^t-\frac {1}{10} e^{K[2]} \left (-5+8 e^{K[2]}\right ) (\cos (2 K[2])+2 \sin (2 K[2]))dK[2]-5 e^t t+3 e^t+8 e^{2 t}-\cos (2 t)-11 \sin (t) \cos (t)\right ) \]

✓ Sympy. Time used: 0.295 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) - 8*exp(2*t) + 5*exp(t) + 4*Derivative(y(t), t) - Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (1 - t\right ) e^{t} + e^{2 t} - \sin {\left (2 t \right )} \]