82.56.2 problem Ex. 2

Internal problem ID [19042]
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. 2
Date solved : Tuesday, January 28, 2025 at 12:46:23 PM
CAS classification : system_of_ODEs

\begin{align*} 4 \frac {d}{d t}x \left (t \right )+9 \frac {d}{d t}y \left (t \right )+2 x \left (t \right )+31 y \left (t \right )&={\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+x \left (t \right )+24 y \left (t \right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 70

dsolve([4*diff(x(t),t)+9*diff(y(t),t)+2*x(t)+31*y(t)=exp(t),3*diff(x(t),t)+7*diff(y(t),t)+x(t)+24*y(t)=3],singsol=all)
 

\begin{align*} x &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} +{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} -\frac {93}{17}+\frac {31 \,{\mathrm e}^{t}}{26} \\ y &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} +{\mathrm e}^{-4 t} \sin \left (t \right ) c_{1} -\frac {2 \,{\mathrm e}^{t}}{13}+\frac {6}{17} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.215 (sec). Leaf size: 79

DSolve[{4*D[x[t],t]+9*D[y[t],t]+2*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+x[t]+24*y[t]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {31 e^t}{26}+c_1 e^{-4 t} \cos (t)-(c_1+c_2) e^{-4 t} \sin (t)-\frac {93}{17} \\ y(t)\to -\frac {2 e^t}{13}+c_2 e^{-4 t} \cos (t)+(2 c_1+c_2) e^{-4 t} \sin (t)+\frac {6}{17} \\ \end{align*}