67.8.2 problem Problem 1(b)

Internal problem ID [14050]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.4 Systems of Linear Differential Equations (Method of Undetermined Coefficients). Problems page 520
Problem number : Problem 1(b)
Date solved : Wednesday, March 05, 2025 at 10:27:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=9 x \left (t \right )-3 y \left (t \right )-6 t\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+11 y \left (t \right )+10 t \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 43

ode:=[diff(x(t),t) = 9*x(t)-3*y(t)-6*t, diff(y(t),t) = -x(t)+11*y(t)+10*t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{8 t} c_{2} +{\mathrm e}^{12 t} c_{1} +\frac {3 t}{8}+\frac {1}{64} \\ y &= \frac {{\mathrm e}^{8 t} c_{2}}{3}-{\mathrm e}^{12 t} c_{1} -\frac {5}{64}-\frac {7 t}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 209

ode={D[x[t],t]==9*x[t]-3*y[t]-6*t,D[y[t],t]==-x[t]+11*y[t]+10*t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{8 t} \left (\left (e^{4 t}+3\right ) \int _1^t3 e^{-12 K[1]} \left (-3+e^{4 K[1]}\right ) K[1]dK[1]-3 \left (e^{4 t}-1\right ) \int _1^te^{-12 K[2]} \left (9+e^{4 K[2]}\right ) K[2]dK[2]+c_1 e^{4 t}-3 c_2 e^{4 t}+3 c_1+3 c_2\right ) \\ y(t)\to \frac {1}{4} e^{8 t} \left (-\left (e^{4 t}-1\right ) \int _1^t3 e^{-12 K[1]} \left (-3+e^{4 K[1]}\right ) K[1]dK[1]+\left (3 e^{4 t}+1\right ) \int _1^te^{-12 K[2]} \left (9+e^{4 K[2]}\right ) K[2]dK[2]+c_1 \left (-e^{4 t}\right )+3 c_2 e^{4 t}+c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*t - 9*x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(-10*t + x(t) - 11*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 3 C_{1} e^{8 t} - C_{2} e^{12 t} + \frac {3 t}{8} + \frac {1}{64}, \ y{\left (t \right )} = C_{1} e^{8 t} + C_{2} e^{12 t} - \frac {7 t}{8} - \frac {5}{64}\right ] \]