46.9.7 problem 7

Internal problem ID [9671]
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
Date solved : Tuesday, September 30, 2025 at 06:25:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+2 y \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+3 y \left (t \right )-{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.201 (sec). Leaf size: 105

ode:=[diff(x(t),t) = 4*x(t)+2*y(t)+exp(t), diff(y(t),t) = -x(t)+3*y(t)-exp(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\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}}{2} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_2}{4}+\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_2}{4}-\frac {{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_1}{4}-\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_1}{4}+\frac {{\mathrm e}^{t}}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.278 (sec). Leaf size: 450

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

\begin{align*} x(t)&\to \frac {1}{7} e^{7 t/2} \left (\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 {5 K[1]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[1]\right )+3 \sqrt {7} \sin \left (\frac {1}{2} \sqrt {7} K[1]\right )\right )dK[1]+4 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {1}{7} e^{-\frac {5 K[2]}{2}} \left (\sqrt {7} \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )-7 \cos \left (\frac {1}{2} \sqrt {7} K[2]\right )\right )dK[2]+7 c_1 \cos \left (\frac {\sqrt {7} t}{2}\right )+\sqrt {7} c_1 \sin \left (\frac {\sqrt {7} t}{2}\right )+4 \sqrt {7} c_2 \sin \left (\frac {\sqrt {7} t}{2}\right )\right )\\ y(t)&\to -\frac {1}{7} e^{7 t/2} \left (2 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right ) \int _1^t\frac {1}{7} e^{-\frac {5 K[1]}{2}} \left (7 \cos \left (\frac {1}{2} \sqrt {7} K[1]\right )+3 \sqrt {7} \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 {5 K[2]}{2}} \left (\sqrt {7} \sin \left (\frac {1}{2} \sqrt {7} K[2]\right )-7 \cos \left (\frac {1}{2} \sqrt {7} K[2]\right )\right )dK[2]-7 c_2 \cos \left (\frac {\sqrt {7} t}{2}\right )+2 \sqrt {7} c_1 \sin \left (\frac {\sqrt {7} t}{2}\right )+\sqrt {7} c_2 \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.296 (sec). Leaf size: 167

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) - 2*y(t) - exp(t) + Derivative(x(t), t),0),Eq(x(t) - 3*y(t) + exp(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + \frac {\sqrt {7} C_{2}}{2}\right ) e^{\frac {7 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + \left (\frac {\sqrt {7} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{\frac {7 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} - \frac {e^{t} \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} - \frac {e^{t} \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2}, \ y{\left (t \right )} = - C_{1} e^{\frac {7 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + C_{2} e^{\frac {7 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {e^{t} \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4} + \frac {e^{t} \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4}\right ] \]