82.55.3 problem Ex. 3

Internal problem ID [19038]
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. 3
Date solved : Tuesday, January 28, 2025 at 12:46:20 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 50

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

\begin{align*} x &= {\mathrm e}^{-5 t} c_{2} +{\mathrm e}^{t} c_{1} +\frac {3 \,{\mathrm e}^{2 t}}{7}-\frac {2 t}{5}-\frac {13}{25} \\ y &= -{\mathrm e}^{-5 t} c_{2} +{\mathrm e}^{t} c_{1} +\frac {4 \,{\mathrm e}^{2 t}}{7}-\frac {12}{25}-\frac {3 t}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.740 (sec). Leaf size: 101

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

\begin{align*} x(t)\to -\frac {2 t}{5}+\frac {3 e^{2 t}}{7}+\frac {1}{2} (c_1-c_2) e^{-5 t}+\frac {1}{2} (c_1+c_2) e^t-\frac {13}{25} \\ y(t)\to -\frac {3}{25} (5 t+4)+\frac {4 e^{2 t}}{7}+\frac {1}{2} (c_2-c_1) e^{-5 t}+\frac {1}{2} (c_1+c_2) e^t \\ \end{align*}