78.5.24 problem 8.c

Internal problem ID [21063]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 8.c
Date solved : Thursday, October 02, 2025 at 07:01:54 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y+\cos \left (t \right )\\ y^{\prime }&=5 x \left (t \right )-2 y+\sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.271 (sec). Leaf size: 36

ode:=[diff(x(t),t) = 2*x(t)-y(t)+cos(t), diff(y(t),t) = 5*x(t)-2*y(t)+sin(t)]; 
ic:=[x(0) = 0, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -\sin \left (t \right )+\cos \left (t \right ) t +\sin \left (t \right ) t \\ y \left (t \right ) &= \cos \left (t \right )-3 \sin \left (t \right )+3 \sin \left (t \right ) t +\cos \left (t \right ) t \\ \end{align*}

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 33

ode={D[x[t],t]==2*x[t]-y[t]+Cos[t],D[y[t],t]==5*x[t]-2*y[t]+Sin[t]}; 
ic={x[0]==0,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to (t-1) \sin (t)+t \cos (t)\\ y(t)&\to 3 (t-1) \sin (t)+(t+1) \cos (t) \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 114

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + y(t) - cos(t) + Derivative(x(t), t),0),Eq(-5*x(t) + 2*y(t) - sin(t) + Derivative(y(t), t),0)] 
ics = {x(0): 0, y(0): 1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {t \sin ^{3}{\left (t \right )}}{5} + \frac {2 t \sin ^{2}{\left (t \right )} \cos {\left (t \right )}}{5} - \frac {t \sin {\left (t \right )} \cos ^{2}{\left (t \right )}}{5} + \frac {6 t \sin {\left (t \right )}}{5} + \frac {2 t \cos ^{3}{\left (t \right )}}{5} + \frac {3 t \cos {\left (t \right )}}{5} - \sin {\left (t \right )}, \ y{\left (t \right )} = t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} + 3 t \sin {\left (t \right )} + t \cos ^{3}{\left (t \right )} - \sin ^{3}{\left (t \right )} - \sin {\left (t \right )} \cos ^{2}{\left (t \right )} - 2 \sin {\left (t \right )} + \cos {\left (t \right )}\right ] \]