61.7.10 problem Problem 5(b)

Internal problem ID [15402]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 5(b)
Date solved : Thursday, October 02, 2025 at 10:12:42 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y+24 \sin \left (t \right )\\ y^{\prime }&=9 x \left (t \right )-3 y+12 \cos \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.273 (sec). Leaf size: 43

ode:=[diff(x(t),t) = 3*x(t)-2*y(t)+24*sin(t), diff(y(t),t) = 9*x(t)-3*y(t)+12*cos(t)]; 
ic:=[x(0) = 1, y(0) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -\frac {4 \sin \left (3 t \right )}{3}+\cos \left (3 t \right )+9 \sin \left (t \right ) \\ y \left (t \right ) &= \frac {7 \cos \left (3 t \right )}{2}-\frac {\sin \left (3 t \right )}{2}-\frac {9 \cos \left (t \right )}{2}+\frac {51 \sin \left (t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 50

ode={D[x[t],t]==3*x[t]-2*y[t]+24*Sin[t],D[y[t],t]==9*x[t]-3*y[t]+12*Cos[t]}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 9 \sin (t)-\frac {4}{3} \sin (3 t)+\cos (3 t)\\ y(t)&\to \frac {1}{2} (51 \sin (t)-\sin (3 t)-9 \cos (t)+7 \cos (3 t)) \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 119

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 2*y(t) - 24*sin(t) + Derivative(x(t), t),0),Eq(-9*x(t) + 3*y(t) - 12*cos(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{3} - \frac {C_{2}}{3}\right ) \cos {\left (3 t \right )} - \left (\frac {C_{1}}{3} + \frac {C_{2}}{3}\right ) \sin {\left (3 t \right )} + 9 \sin {\left (t \right )} \sin ^{2}{\left (3 t \right )} + 9 \sin {\left (t \right )} \cos ^{2}{\left (3 t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (3 t \right )} - C_{2} \sin {\left (3 t \right )} + \frac {51 \sin {\left (t \right )} \sin ^{2}{\left (3 t \right )}}{2} + \frac {51 \sin {\left (t \right )} \cos ^{2}{\left (3 t \right )}}{2} - \frac {9 \sin ^{2}{\left (3 t \right )} \cos {\left (t \right )}}{2} - \frac {9 \cos {\left (t \right )} \cos ^{2}{\left (3 t \right )}}{2}\right ] \]