61.6.7 problem Problem 4(g)

Internal problem ID [15392]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6.4 Reduction to a single ODE. Problems page 415
Problem number : Problem 4(g)
Date solved : Thursday, October 02, 2025 at 10:12:30 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.245 (sec). Leaf size: 50

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

\begin{align*} x \left (t \right ) &= -8 \,{\mathrm e}^{-\frac {t}{8}} c_1 -\frac {6 \sin \left (t \right )}{65}-\frac {17 \cos \left (t \right )}{65}+2 t +c_2 \\ y \left (t \right ) &= 12 \,{\mathrm e}^{-\frac {t}{8}} c_1 -\frac {7 \cos \left (t \right )}{65}+\frac {9 \sin \left (t \right )}{65}+8-3 t -c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 266

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

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

✓ Sympy. Time used: 0.138 (sec). Leaf size: 61

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

\[ \left [ x{\left (t \right )} = - C_{1} - \frac {2 C_{2} e^{- \frac {t}{8}}}{3} + 2 t - \frac {6 \sin {\left (t \right )}}{65} - \frac {17 \cos {\left (t \right )}}{65} - 16, \ y{\left (t \right )} = C_{1} + C_{2} e^{- \frac {t}{8}} - 3 t + \frac {9 \sin {\left (t \right )}}{65} - \frac {7 \cos {\left (t \right )}}{65} + 24\right ] \]