Internal problem ID [5788]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page
400
Problem number: 4(c).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=-4 x \relax (t )+y \relax (t )-t +3\\ y^{\prime }\relax (t )&=-x \relax (t )-5 y \relax (t )+t +1 \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 106
dsolve([diff(x(t),t)=-4*x(t)+y(t)-t+3,diff(y(t),t)=-x(t)-5*y(t)+t+1],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = -\frac {{\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}+\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}}{2}+\frac {39}{49}-\frac {4 t}{21} \] \[ y \relax (t ) = {\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}+{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1}+\frac {5 t}{21}-\frac {1}{147} \]
✓ Solution by Mathematica
Time used: 0.486 (sec). Leaf size: 359
DSolve[{x'[t]==-4*x[t]+y[t]-t+3,y'[t]==-x[t]-5*y[t]+t+1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{441} e^{-9 t/2} \left (147 \left (\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+3 \cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \int _1^t-\frac {1}{3} e^{\frac {9 K[1]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) (K[1]-3)+\sqrt {3} (K[1]+5) \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )dK[1]+e^{9 t/2} \left (-21 t+\sqrt {3} (35 t-1) \sin \left (\sqrt {3} t\right )+(21 t-233) \cos \left (\sqrt {3} t\right )+233\right )+147 \left (3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1+2 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right )\right ) \\ y(t)\to \frac {1}{441} \left (-147 \sqrt {3} e^{-9 t/2} \sin \left (\frac {\sqrt {3} t}{2}\right ) \left (2 \int _1^t-\frac {1}{3} e^{\frac {9 K[1]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) (K[1]-3)+\sqrt {3} (K[1]+5) \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )dK[1]+2 c_1+c_2\right )+63 t+\sqrt {3} (117-28 t) \sin \left (\sqrt {3} t\right )+(42 t+115) \cos \left (\sqrt {3} t\right )+441 c_2 e^{-9 t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )-118\right ) \\ \end{align*}