67.8.1 problem Problem 1(a)

Internal problem ID [14128]
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 : Tuesday, January 28, 2025 at 06:15:24 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*}

✓ Solution by Maple

Time used: 1.106 (sec). Leaf size: 135

dsolve([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)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{6 t}+{\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 &= -\frac {2 \cosh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+\frac {2 \sinh \left (7 t \right ) {\mathrm e}^{6 t}}{7}+c_{2} {\mathrm e}^{6 t}-\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*}

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 108

DSolve[{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]},{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*}