50.28.9 problem 5(b)

Internal problem ID [8196]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section 10.3 Homogeneous Linear Systems with Constant Coefficients. Page 387
Problem number : 5(b)
Date solved : Monday, January 27, 2025 at 03:45:40 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-5 t +2\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-2 y \left (t \right )-8 t -8 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 42

dsolve([diff(x(t),t)=x(t)+y(t)-5*t+2,diff(y(t),t)=4*x(t)-2*y(t)-8*t-8],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+c_{1} {\mathrm e}^{2 t}+3 t +2 \\ y &= -4 c_{2} {\mathrm e}^{-3 t}+c_{1} {\mathrm e}^{2 t}-1+2 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.194 (sec). Leaf size: 92

DSolve[{D[x[t],t]==x[t]+y[t]-5*t+2,D[y[t],t]==4*x[t]-2*y[t]-8*t-8},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-3 t} \left (5 e^{3 t} (3 t+2)+(4 c_1+c_2) e^{5 t}+c_1-c_2\right ) \\ y(t)\to \frac {1}{5} e^{-3 t} \left (5 e^{3 t} (2 t-1)+(4 c_1+c_2) e^{5 t}-4 c_1+4 c_2\right ) \\ \end{align*}