28.4.23 problem 7.23

Internal problem ID [4555]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.23
Date solved : Monday, January 27, 2025 at 09:23:26 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = x_{0}\\ y \left (0\right ) = y_{0} \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 87

dsolve([diff(x(t),t)-2*x(t)-y(t) = 2*exp(t), x(t)-diff(y(t),t)+2*y(t) = 3*exp(4*t), x(0) = x__0, y(0) = y__0], singsol=all)
 

\begin{align*} x &= \left (-\frac {3}{2}+\frac {x_{0}}{2}-\frac {y_{0}}{2}\right ) {\mathrm e}^{t}+\left (2+\frac {x_{0}}{2}+\frac {y_{0}}{2}\right ) {\mathrm e}^{3 t}-{\mathrm e}^{4 t}+t \,{\mathrm e}^{t}+\frac {{\mathrm e}^{t}}{2} \\ y &= -\left (-\frac {3}{2}+\frac {x_{0}}{2}-\frac {y_{0}}{2}\right ) {\mathrm e}^{t}+\left (2+\frac {x_{0}}{2}+\frac {y_{0}}{2}\right ) {\mathrm e}^{3 t}-2 \,{\mathrm e}^{4 t}-\frac {3 \,{\mathrm e}^{t}}{2}-t \,{\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.097 (sec). Leaf size: 73

DSolve[{D[x[t],t]-2*x[t]-y[t]==2*Exp[t],x[t]-D[y[t],t]+2*y[t]==3*Exp[4*t]},{x[0]==x0,y[0]==y0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^t \left (e^{2 t} (\text {x0}+\text {y0}+4)+2 t-2 e^{3 t}+\text {x0}-\text {y0}-2\right ) \\ y(t)\to \frac {1}{2} e^t \left (e^{2 t} (\text {x0}+\text {y0}+4)-2 t-4 e^{3 t}-\text {x0}+\text {y0}\right ) \\ \end{align*}