Internal problem ID [15543]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 3 (Systems of differential equations). Section 23.2 The method of undetermined
coefficients. Exercises page 239
Problem number: 820.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )-x \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )+{\mathrm e}^{t} \end {align*}
With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 1] \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 28
dsolve([diff(x(t),t) = y(t)-x(t)+exp(t), diff(y(t),t) = x(t)-y(t)+exp(t), x(0) = 0, y(0) = 1], singsol=all)
\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{-2 t}}{2}+{\mathrm e}^{t}-\frac {1}{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-2 t}}{2}+{\mathrm e}^{t}-\frac {1}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 40
DSolve[{x'[t]==y[t]-x[t]+Exp[t],y'[t]==x[t]-y[t]+Exp[t]},{x[0]==0,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {e^{-2 t}}{2}+e^t-\frac {1}{2} \\ y(t)\to \frac {e^{-2 t}}{2}+e^t-\frac {1}{2} \\ \end{align*}