61.7.5 problem Problem 4(a)

Internal problem ID [15397]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 4(a)
Date solved : Thursday, October 02, 2025 at 10:12:36 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+4 y-2 z \left (t \right )-2 \sinh \left (t \right )\\ y^{\prime }&=4 x \left (t \right )+2 y-2 z \left (t \right )+10 \cosh \left (t \right )\\ z^{\prime }\left (t \right )&=-x \left (t \right )+3 y+z \left (t \right )+5 \end{align*}

✓ Maple. Time used: 0.451 (sec). Leaf size: 362

ode:=[diff(x(t),t) = 2*x(t)+4*y(t)-2*z(t)-2*sinh(t), diff(y(t),t) = 4*x(t)+2*y(t)-2*z(t)+10*cosh(t), diff(z(t),t) = -x(t)+3*y(t)+z(t)+5]; 
dsolve(ode);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 233

ode={D[x[t],t]==2*x[t]+4*y[t]-2*z[t]-2*Sinh[t],D[y[t],t]==4*x[t]+2*y[t]-2*z[t]+10*Cosh[t],D[z[t],t]==-x[t]+3*y[t]+z[t]+5}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {29 e^{-t}}{9}-3 e^t+\frac {9}{14} (c_1-c_2) e^{-2 t}+\frac {2}{21} (9 c_1+5 c_2-7 c_3) e^{5 t}+\frac {1}{6} (-3 c_1+c_2+4 c_3) e^{2 t}-1\\ y(t)&\to \frac {7 e^{-t}}{9}-e^t+\frac {5}{14} (c_2-c_1) e^{-2 t}+\frac {2}{21} (9 c_1+5 c_2-7 c_3) e^{5 t}+\frac {1}{6} (-3 c_1+c_2+4 c_3) e^{2 t}-1\\ z(t)&\to -\frac {25 e^{-t}}{9}-4 e^t+\frac {4}{7} (c_1-c_2) e^{-2 t}+\frac {1}{21} (9 c_1+5 c_2-7 c_3) e^{5 t}+\frac {1}{3} (-3 c_1+c_2+4 c_3) e^{2 t}-3 \end{align*}

✓ Sympy. Time used: 0.353 (sec). Leaf size: 126

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) - 4*y(t) + 2*z(t) + 2*sinh(t) + Derivative(x(t), t),0),Eq(-4*x(t) - 2*y(t) + 2*z(t) - 10*cosh(t) + Derivative(y(t), t),0),Eq(x(t) - 3*y(t) - z(t) + Derivative(z(t), t) - 5,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {9 C_{1} e^{- 2 t}}{8} + \frac {C_{2} e^{2 t}}{2} + 2 C_{3} e^{5 t} + \frac {2 \sinh {\left (t \right )}}{9} - \frac {56 \cosh {\left (t \right )}}{9} - 1, \ y{\left (t \right )} = - \frac {5 C_{1} e^{- 2 t}}{8} + \frac {C_{2} e^{2 t}}{2} + 2 C_{3} e^{5 t} - \frac {16 \sinh {\left (t \right )}}{9} - \frac {2 \cosh {\left (t \right )}}{9} - 1, \ z{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{2 t} + C_{3} e^{5 t} - \frac {11 \sinh {\left (t \right )}}{9} - \frac {61 \cosh {\left (t \right )}}{9} - 3\right ] \]