67.27.15 problem 38.10 (i)

Internal problem ID [17058]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (i)
Date solved : Thursday, October 02, 2025 at 01:42:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+3 y \left (t \right )-6 \,{\mathrm e}^{3 t}\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+6 y \left (t \right )+2 \,{\mathrm e}^{3 t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.174 (sec). Leaf size: 43

ode:=[diff(x(t),t) = 4*x(t)+3*y(t)-6*exp(3*t), diff(y(t),t) = x(t)+6*y(t)+2*exp(3*t)]; 
ic:=[x(0) = 4, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{3 t}+{\mathrm e}^{7 t}-6 t \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -{\mathrm e}^{3 t}+{\mathrm e}^{7 t}+2 t \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 50

ode={D[x[t],t]==4*x[t]+3*y[t]+6*Exp[3*t],D[y[t],t]==x[t]+6*y[t]+2*Exp[3*t]}; 
ic={x[0]==4,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{3 t} \left (12 t+7 e^{4 t}+9\right )\\ y(t)&\to \frac {1}{4} e^{3 t} \left (-4 t+7 e^{4 t}-7\right ) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) - 3*y(t) + 6*exp(3*t) + Derivative(x(t), t),0),Eq(-x(t) - 6*y(t) - 2*exp(3*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 C_{1} e^{3 t} + C_{2} e^{7 t} - 6 t e^{3 t}, \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{7 t} + 2 t e^{3 t}\right ] \]