61.7.4 problem Problem 3(d)

Internal problem ID [15396]
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 3(d)
Date solved : Thursday, October 02, 2025 at 10:12:35 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-14 x \left (t \right )+39 y+78 \sinh \left (t \right )\\ y^{\prime }&=-6 x \left (t \right )+16 y+6 \cosh \left (t \right ) \end{align*}

✓ Maple. Time used: 0.462 (sec). Leaf size: 85

ode:=[diff(x(t),t) = -14*x(t)+39*y(t)+78*sinh(t), diff(y(t),t) = -6*x(t)+16*y(t)+6*cosh(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \sin \left (3 t \right ) c_2 +{\mathrm e}^{t} \cos \left (3 t \right ) c_1 +60 \,{\mathrm e}^{-t}-52 \,{\mathrm e}^{t} \\ y \left (t \right ) &= \frac {5 \,{\mathrm e}^{t} \sin \left (3 t \right ) c_2}{13}+\frac {{\mathrm e}^{t} \cos \left (3 t \right ) c_2}{13}+\frac {5 \,{\mathrm e}^{t} \cos \left (3 t \right ) c_1}{13}-\frac {{\mathrm e}^{t} \sin \left (3 t \right ) c_1}{13}+20 \,{\mathrm e}^{-t}-20 \,{\mathrm e}^{t}-2 \sinh \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 9.246 (sec). Leaf size: 1975

ode={D[x[t],t]==-14*x[t]+39*y[t]+78*Sinh[t],D[y[t],t]==-6*x[t]+16*y[t]+6*Cosh[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 0.493 (sec). Leaf size: 129

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(14*x(t) - 39*y(t) - 78*sinh(t) + Derivative(x(t), t),0),Eq(6*x(t) - 16*y(t) - 6*cosh(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + \frac {5 C_{2}}{2}\right ) e^{t} \cos {\left (3 t \right )} - \left (\frac {5 C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{t} \sin {\left (3 t \right )} - 112 \sin ^{2}{\left (3 t \right )} \sinh {\left (t \right )} + 8 \sin ^{2}{\left (3 t \right )} \cosh {\left (t \right )} - 112 \cos ^{2}{\left (3 t \right )} \sinh {\left (t \right )} + 8 \cos ^{2}{\left (3 t \right )} \cosh {\left (t \right )}, \ y{\left (t \right )} = - C_{1} e^{t} \sin {\left (3 t \right )} + C_{2} e^{t} \cos {\left (3 t \right )} - 42 \sin ^{2}{\left (3 t \right )} \sinh {\left (t \right )} - 42 \cos ^{2}{\left (3 t \right )} \sinh {\left (t \right )}\right ] \]