Internal problem ID [5964]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
EXERCISES 8.1. Page 332
Problem number: 7.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=4 x \relax (t )+2 y \relax (t )+{\mathrm e}^{t}\\ y^{\prime }\relax (t )&=-x \relax (t )+3 y \relax (t )-{\mathrm e}^{t} \end {align*}
✓ Solution by Maple
Time used: 0.101 (sec). Leaf size: 106
dsolve([diff(x(t),t)=4*x(t)+2*y(t)+exp(t),diff(y(t),t)=-x(t)+3*y(t)-exp(t)],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = -\frac {{\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{2}+\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{2}-\frac {{\mathrm e}^{t}}{2} \] \[ y \relax (t ) = {\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}+{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}+\frac {{\mathrm e}^{t}}{4} \]
✓ Solution by Mathematica
Time used: 0.183 (sec). Leaf size: 127
DSolve[{x'[t]==4*x[t]+2*y[t]+Exp[t],y'[t]==-x[t]+3*y[t]-Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {e^t}{2}+\frac {1}{7} e^{7 t/2} \left (7 c_1 \cos \left (\frac {\sqrt {7} t}{2}\right )+\sqrt {7} (c_1+4 c_2) \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \\ y(t)\to \frac {e^t}{4}+\frac {1}{7} e^{7 t/2} \left (7 c_2 \cos \left (\frac {\sqrt {7} t}{2}\right )-\sqrt {7} (2 c_1+c_2) \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \\ \end{align*}