14.20.10 problem 10

Internal problem ID [2707]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 10
Date solved : Monday, January 27, 2025 at 06:12:06 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 31

dsolve([diff(x(t),t) = 3*x(t)-4*y(t)+exp(t), diff(y(t),t) = x(t)-y(t)+exp(t), x(0) = 1, y(0) = 1], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 31

DSolve[{D[x[t],t]==3*x[t]-4*y[t]+Exp[t],D[y[t],t]==x[t]-y[t]+Exp[t]},{x[0]==1,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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