82.56.3 problem Ex. 4

Internal problem ID [19043]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter XI. Ordinary differential equations with more than two variables. End of chapter problems at page 143
Problem number : Ex. 4
Date solved : Tuesday, January 28, 2025 at 12:46:24 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+4 x \left (t \right )+3 y \left (t \right )&=t\\ \frac {d}{d t}y \left (t \right )+2 x \left (t \right )+5 y \left (t \right )&={\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 50

dsolve([diff(x(t),t)+4*x(t)+3*y(t)=t,diff(y(t),t)+2*x(t)+5*y(t)=exp(t)],singsol=all)
 

\begin{align*} x &= c_{2} {\mathrm e}^{-2 t}+{\mathrm e}^{-7 t} c_{1} -\frac {31}{196}-\frac {{\mathrm e}^{t}}{8}+\frac {5 t}{14} \\ y &= -\frac {2 c_{2} {\mathrm e}^{-2 t}}{3}+{\mathrm e}^{-7 t} c_{1} +\frac {5 \,{\mathrm e}^{t}}{24}+\frac {9}{98}-\frac {t}{7} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.192 (sec). Leaf size: 109

DSolve[{D[x[t],t]+4*x[t]+3*y[t]==t,D[y[t],t]+2*x[t]+5*y[t]==Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{196} (70 t-31)-\frac {e^t}{8}+\frac {3}{5} (c_1-c_2) e^{-2 t}+\frac {1}{5} (2 c_1+3 c_2) e^{-7 t} \\ y(t)\to -\frac {t}{7}+\frac {5 e^t}{24}-\frac {2}{5} (c_1-c_2) e^{-2 t}+\frac {1}{5} (2 c_1+3 c_2) e^{-7 t}+\frac {9}{98} \\ \end{align*}