50.29.11 problem 4(a)

Internal problem ID [8207]
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)
Date solved : Monday, January 27, 2025 at 03:45:59 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )-4 t +1\\ y^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right )+3 t +4 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 105

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],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\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 {7 t}{2}+\frac {25}{8} \\ y &= \frac {{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {5}{16}+\frac {t}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.657 (sec). Leaf size: 128

DSolve[{D[x[t],t]==x[t]+2*y[t]-4+t+1,D[y[t],t]==-x[t]+2*y[t]+3*t+4},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to t+c_1 e^{3 t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )-\frac {(c_1-4 c_2) e^{3 t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )}{\sqrt {7}}+\frac {9}{2} \\ y(t)\to -t+c_2 e^{3 t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )-\frac {(2 c_1-c_2) e^{3 t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )}{\sqrt {7}}-\frac {1}{4} \\ \end{align*}