61.8.1 problem Problem 1(a)

Internal problem ID [15409]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.4 Systems of Linear Differential Equations (Method of Undetermined Coefficients). Problems page 520
Problem number : Problem 1(a)
Date solved : Thursday, October 02, 2025 at 10:13:47 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+5 y+10 \sinh \left (t \right )\\ y^{\prime }&=19 x \left (t \right )-13 y+24 \sinh \left (t \right ) \end{align*}

✓ Maple. Time used: 0.488 (sec). Leaf size: 135

ode:=[diff(x(t),t) = x(t)+5*y(t)+10*sinh(t), diff(y(t),t) = 19*x(t)-13*y(t)+24*sinh(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{6 t} c_2 +{\mathrm e}^{-18 t} c_1 +\frac {5 \,{\mathrm e}^{-18 t} \left (\left (-\frac {221 \cosh \left (5 t \right )}{60}+\frac {17 \cosh \left (7 t \right )}{7}+\frac {221 \sinh \left (5 t \right )}{60}-\frac {17 \sinh \left (7 t \right )}{7}\right ) {\mathrm e}^{24 t}+\sinh \left (17 t \right )-\frac {221 \sinh \left (19 t \right )}{228}+\cosh \left (17 t \right )-\frac {221 \cosh \left (19 t \right )}{228}\right )}{17} \\ y \left (t \right ) &= -\frac {2 \cosh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+\frac {2 \sinh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+{\mathrm e}^{6 t} c_2 -\frac {2 \,{\mathrm e}^{-18 t} \sinh \left (17 t \right )}{17}-\frac {2 \,{\mathrm e}^{-18 t} \cosh \left (17 t \right )}{17}-\frac {19 \,{\mathrm e}^{-18 t} c_1}{5}-2 \sinh \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 108

ode={D[x[t],t]==x[t]+5*y[t]+10*Sinh[t],D[y[t],t]==19*x[t]-13*y[t]+24*Sinh[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {120 e^{-t}}{119}-\frac {26 e^t}{19}+\frac {5}{24} (c_1-c_2) e^{-18 t}+\frac {1}{24} (19 c_1+5 c_2) e^{6 t}\\ y(t)&\to \frac {71 e^{-t}}{119}-e^t-\frac {19}{24} (c_1-c_2) e^{-18 t}+\frac {1}{24} (19 c_1+5 c_2) e^{6 t} \end{align*}

✓ Sympy. Time used: 0.204 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 5*y(t) - 10*sinh(t) + Derivative(x(t), t),0),Eq(-19*x(t) + 13*y(t) - 24*sinh(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {5 C_{1} e^{- 18 t}}{19} + C_{2} e^{6 t} - \frac {5374 \sinh {\left (t \right )}}{2261} - \frac {814 \cosh {\left (t \right )}}{2261}, \ y{\left (t \right )} = C_{1} e^{- 18 t} + C_{2} e^{6 t} - \frac {190 \sinh {\left (t \right )}}{119} - \frac {48 \cosh {\left (t \right )}}{119}\right ] \]