72.9.14 problem 29

Internal problem ID [14810]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 29
Date solved : Tuesday, January 28, 2025 at 07:17:04 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+3 y\\ y^{\prime }&=x \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 33

dsolve([diff(x(t),t) = 2*x(t)+3*y(t), diff(y(t),t) = x(t), x(0) = 2, y(0) = 3], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {15 \,{\mathrm e}^{3 t}}{4}-\frac {7 \,{\mathrm e}^{-t}}{4} \\ y &= \frac {5 \,{\mathrm e}^{3 t}}{4}+\frac {7 \,{\mathrm e}^{-t}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 44

DSolve[{D[x[t],t]==2*x[t]+3*y[t],D[y[t],t]==x[t]},{x[0]==2,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (15 e^{4 t}-7\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (5 e^{4 t}+7\right ) \\ \end{align*}