Internal problem ID [5786]
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(a).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )+2 y \relax (t )-4 t +1\\ y^{\prime }\relax (t )&=-x \relax (t )+2 y \relax (t )+3 t +4 \end {align*}
✓ Solution by Maple
Time used: 0.066 (sec). Leaf size: 106
dsolve([diff(x(t),t)=x(t)+2*y(t)-4*t+1,diff(y(t),t)=-x(t)+2*y(t)+3*t+4],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = \frac {{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{2}+\frac {{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{2}+\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{2}+\frac {25}{8}+\frac {7 t}{2} \] \[ y \relax (t ) = {\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}+{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}+\frac {t}{4}-\frac {5}{16} \]
✓ Solution by Mathematica
Time used: 0.656 (sec). Leaf size: 449
DSolve[{x'[t]==x[t]+2*y[t]-4+t+1,y'[t]==-x[t]+2*y[t]+3*t+4},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{7} e^{3 t/2} \left (\left (7 \cos \left (\frac {\sqrt {7} t}{2}\right )-\sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[1]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[1]\right ) (K[1]-3)-\sqrt {7} (11 K[1]+19) \sin \left (\frac {1}{2} \sqrt {7} K[1]\right )\right )dK[1]+\sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right ) \left (4 \int _1^t\frac {1}{7} e^{-\frac {3 K[2]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[2]\right ) (3 K[2]+4)-\sqrt {7} (K[2]+10) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )\right )dK[2]-c_1+4 c_2\right )+7 c_1 \cos \left (\frac {\sqrt {7} t}{2}\right )\right ) \\ y(t)\to \frac {1}{7} e^{3 t/2} \left (-2 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[1]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[1]\right ) (K[1]-3)-\sqrt {7} (11 K[1]+19) \sin \left (\frac {1}{2} \sqrt {7} K[1]\right )\right )dK[1]+\left (\sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )+7 \cos \left (\frac {\sqrt {7} t}{2}\right )\right ) \int _1^t\frac {1}{7} e^{-\frac {3 K[2]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[2]\right ) (3 K[2]+4)-\sqrt {7} (K[2]+10) \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )\right )dK[2]+7 c_2 \cos \left (\frac {\sqrt {7} t}{2}\right )+\sqrt {7} (c_2-2 c_1) \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \\ \end{align*}