55.1.15 problem HW 5 problem 7

Internal problem ID [8711]
Book : Selected problems from homeworks from different courses
Section : Math 2520, summer 2021. Differential Equations and Linear Algebra. Normandale college, Bloomington, Minnesota
Problem number : HW 5 problem 7
Date solved : Monday, January 27, 2025 at 04:20:02 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 74

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

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