Internal problem ID [12209]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 3, SYSTEMS OF DIFFERENTIAL EQUATIONS. Problems page
209
Problem number: Problem 3.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-5 x \left (t \right )-y \left (t \right )+{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{2 t} \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 102
dsolve([diff(x(t),t)+5*x(t)+y(t)=exp(t),diff(y(t),t)-x(t)-3*y(t)=exp(2*t)],singsol=all)
\begin{align*} x \left (t \right ) &= {\mathrm e}^{\left (\sqrt {15}-1\right ) t} c_{2} +{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_{1} +\frac {{\mathrm e}^{2 t}}{6}+\frac {2 \,{\mathrm e}^{t}}{11} \\ y \left (t \right ) &= -{\mathrm e}^{\left (\sqrt {15}-1\right ) t} c_{2} \sqrt {15}+{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_{1} \sqrt {15}-4 \,{\mathrm e}^{\left (\sqrt {15}-1\right ) t} c_{2} -4 \,{\mathrm e}^{-\left (1+\sqrt {15}\right ) t} c_{1} -\frac {{\mathrm e}^{t}}{11}-\frac {7 \,{\mathrm e}^{2 t}}{6} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.39 (sec). Leaf size: 206
DSolve[{x'[t]+5*x[t]+y[t]==Exp[t],y'[t]-x[t]-3*y[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{330} e^{-\left (\left (1+\sqrt {15}\right ) t\right )} \left (60 e^{\left (2+\sqrt {15}\right ) t}+55 e^{\left (3+\sqrt {15}\right ) t}-11 \left (\left (4 \sqrt {15}-15\right ) c_1+\sqrt {15} c_2\right ) e^{2 \sqrt {15} t}+11 \left (\left (15+4 \sqrt {15}\right ) c_1+\sqrt {15} c_2\right )\right ) \\ y(t)\to -\frac {1}{330} e^{-\left (\left (1+\sqrt {15}\right ) t\right )} \left (30 e^{\left (2+\sqrt {15}\right ) t}+385 e^{\left (3+\sqrt {15}\right ) t}-11 \left (\sqrt {15} c_1+\left (15+4 \sqrt {15}\right ) c_2\right ) e^{2 \sqrt {15} t}+11 \left (\sqrt {15} c_1+\left (4 \sqrt {15}-15\right ) c_2\right )\right ) \\ \end{align*}