61.7.6 problem Problem 4(b)

Internal problem ID [15398]
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(b)
Date solved : Thursday, October 02, 2025 at 10:12:38 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+6 y-2 z \left (t \right )+50 \,{\mathrm e}^{t}\\ y^{\prime }&=6 x \left (t \right )+2 y-2 z \left (t \right )+21 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=-x \left (t \right )+6 y+z \left (t \right )+9 \end{align*}

✓ Maple. Time used: 0.203 (sec). Leaf size: 101

ode:=[diff(x(t),t) = 2*x(t)+6*y(t)-2*z(t)+50*exp(t), diff(y(t),t) = 6*x(t)+2*y(t)-2*z(t)+21*exp(-t), diff(z(t),t) = -x(t)+6*y(t)+z(t)+9]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-4 t}+\frac {2 c_2 \,{\mathrm e}^{3 t}}{5}+c_3 \,{\mathrm e}^{6 t}-6 \,{\mathrm e}^{-t}+12 \,{\mathrm e}^{t}-1 \\ y \left (t \right ) &= -\frac {2 c_1 \,{\mathrm e}^{-4 t}}{3}+\frac {2 c_2 \,{\mathrm e}^{3 t}}{5}+c_3 \,{\mathrm e}^{6 t}+{\mathrm e}^{-t}+2 \,{\mathrm e}^{t}-1 \\ z \left (t \right ) &= -6 \,{\mathrm e}^{-t}-4+37 \,{\mathrm e}^{t}+c_1 \,{\mathrm e}^{-4 t}+c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t} \\ \end{align*}

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 213

ode={D[x[t],t]==2*x[t]+6*y[t]-2*z[t]+50*Exp[t],D[y[t],t]==6*x[t]+2*y[t]-2*z[t]+21*Exp[-t],D[z[t],t]==-x[t]+6*y[t]+z[t]+9}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -6 e^{-t}+12 e^t+\frac {3}{5} (c_1-c_2) e^{-4 t}+\frac {1}{15} (16 c_1+9 c_2-10 c_3) e^{6 t}-\frac {2}{3} (c_1-c_3) e^{3 t}-1\\ y(t)&\to e^{-t}+2 e^t-\frac {2}{5} (c_1-c_2) e^{-4 t}+\frac {1}{15} (16 c_1+9 c_2-10 c_3) e^{6 t}-\frac {2}{3} (c_1-c_3) e^{3 t}-1\\ z(t)&\to -6 e^{-t}+37 e^t+\frac {3}{5} (c_1-c_2) e^{-4 t}+\frac {1}{15} (16 c_1+9 c_2-10 c_3) e^{6 t}-\frac {5}{3} (c_1-c_3) e^{3 t}-4 \end{align*}

✓ Sympy. Time used: 0.237 (sec). Leaf size: 112

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) - 6*y(t) + 2*z(t) - 50*exp(t) + Derivative(x(t), t),0),Eq(-6*x(t) - 2*y(t) + 2*z(t) + Derivative(y(t), t) - 21*exp(-t),0),Eq(x(t) - 6*y(t) - z(t) + Derivative(z(t), t) - 9,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- 4 t} + \frac {2 C_{2} e^{3 t}}{5} + C_{3} e^{6 t} + 12 e^{t} - 1 - 6 e^{- t}, \ y{\left (t \right )} = - \frac {2 C_{1} e^{- 4 t}}{3} + \frac {2 C_{2} e^{3 t}}{5} + C_{3} e^{6 t} + 2 e^{t} - 1 + e^{- t}, \ z{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{3 t} + C_{3} e^{6 t} + 37 e^{t} - 4 - 6 e^{- t}\right ] \]