82.56.1 problem Ex. 1

Internal problem ID [19041]
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. 1
Date solved : Tuesday, January 28, 2025 at 12:46:22 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 42

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

\begin{align*} x &= -\frac {3 \,{\mathrm e}^{t}}{11}+\frac {{\mathrm e}^{2 t}}{2}+{\mathrm e}^{-\frac {6 t}{5}} c_{2} \\ y &= \frac {15 \,{\mathrm e}^{t}}{22}-8 \,{\mathrm e}^{-\frac {6 t}{5}} c_{2} +{\mathrm e}^{-t} c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.323 (sec). Leaf size: 65

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

\begin{align*} x(t)\to \frac {1}{22} e^t \left (11 e^t-6\right )+c_1 e^{-6 t/5} \\ y(t)\to \frac {15 e^t}{22}-8 c_1 e^{-6 t/5}+(8 c_1+c_2) e^{-t} \\ \end{align*}