1.23.11 problem 11

Internal problem ID [597]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 04:01:46 AM
CAS classification : system_of_ODEs

\begin{align*} -\frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )&=x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )-4 \frac {d}{d t}y \left (t \right )&=x \left (t \right )-15 y \left (t \right )+{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.295 (sec). Leaf size: 63

ode:=[-diff(x(t),t)+2*diff(y(t),t) = x(t)+3*y(t)+exp(t), 3*diff(x(t),t)-4*diff(y(t),t) = x(t)-15*y(t)+exp(-t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \sin \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_1 -\frac {11 \,{\mathrm e}^{t}}{20}-\frac {{\mathrm e}^{-t}}{4} \\ y \left (t \right ) &= \frac {\cos \left (3 t \right ) c_1}{3}-\frac {\cos \left (3 t \right ) c_2}{3}+\frac {\sin \left (3 t \right ) c_1}{3}+\frac {\sin \left (3 t \right ) c_2}{3}+\frac {{\mathrm e}^{t}}{10} \\ \end{align*}

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 77

ode={-D[x[t],t]+2*D[y[t],t]==x[t]+3*y[t]+Exp[t],3*D[x[t],t]-4*D[y[t],t]==x[t]-15*y[t]+Exp[-t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {1}{20} e^{-t} \left (11 e^{2 t}+5\right )+c_1 \cos (3 t)+(c_1-3 c_2) \sin (3 t)\\ y(t)&\to \frac {e^t}{10}+c_2 \cos (3 t)+\left (\frac {2 c_1}{3}-c_2\right ) \sin (3 t) \end{align*}

✓ Sympy. Time used: 0.281 (sec). Leaf size: 122

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 3*y(t) - exp(t) - Derivative(x(t), t) + 2*Derivative(y(t), t),0),Eq(-x(t) + 15*y(t) + 3*Derivative(x(t), t) - 4*Derivative(y(t), t) - exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {3 C_{1}}{2} - \frac {3 C_{2}}{2}\right ) \cos {\left (3 t \right )} - \left (\frac {3 C_{1}}{2} + \frac {3 C_{2}}{2}\right ) \sin {\left (3 t \right )} - \frac {11 e^{t} \sin ^{2}{\left (3 t \right )}}{20} - \frac {11 e^{t} \cos ^{2}{\left (3 t \right )}}{20} - \frac {e^{- t} \sin ^{2}{\left (3 t \right )}}{4} - \frac {e^{- t} \cos ^{2}{\left (3 t \right )}}{4}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )} + \frac {e^{t} \sin ^{2}{\left (3 t \right )}}{10} + \frac {e^{t} \cos ^{2}{\left (3 t \right )}}{10}\right ] \]