67.8.2 problem Problem 1(b)

Internal problem ID [14129]
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 : Tuesday, January 28, 2025 at 06:15:25 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

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

dsolve([diff(x(t),t)=9*x(t)-3*y(t)-6*t,diff(y(t),t)=-x(t)+11*y(t)+10*t],singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==9*x[t]-3*y[t]-6*t,D[y[t],t]==-x[t]+11*y[t]+10*t},{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*}