82.55.4 problem Ex. 4

Internal problem ID [19039]
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. problems at page 129
Problem number : Ex. 4
Date solved : Tuesday, January 28, 2025 at 12:46:21 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 )+44 x \left (t \right )+49 y \left (t \right )&=t\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+34 x \left (t \right )+38 y \left (t \right )&={\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 51

dsolve([4*diff(x(t),t)+9*diff(y(t),t)+44*x(t)+49*y(t)=t,3*diff(x(t),t)+7*diff(y(t),t)+34*x(t)+38*y(t)=exp(t)],singsol=all)
 

\begin{align*} x &= {\mathrm e}^{-t} c_{2} +c_{1} {\mathrm e}^{-6 t}-\frac {29 \,{\mathrm e}^{t}}{7}+\frac {19 t}{3}-\frac {56}{9} \\ y &= -{\mathrm e}^{-t} c_{2} +4 c_{1} {\mathrm e}^{-6 t}+\frac {24 \,{\mathrm e}^{t}}{7}+\frac {55}{9}-\frac {17 t}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.162 (sec). Leaf size: 104

DSolve[{4*D[x[t],t]+9*D[y[t],t]+44*x[t]+49*y[t]==t,3*D[x[t],t]+7*D[y[t],t]+34*x[t]+38*y[t]==Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{9} (57 t-56)-\frac {29 e^t}{7}+\frac {1}{5} (4 c_1-c_2) e^{-t}+\frac {1}{5} (c_1+c_2) e^{-6 t} \\ y(t)\to \frac {1}{9} (55-51 t)+\frac {24 e^t}{7}+\frac {1}{5} (c_2-4 c_1) e^{-t}+\frac {4}{5} (c_1+c_2) e^{-6 t} \\ \end{align*}