12.22.14 problem section 10.5, problem 14

Internal problem ID [2267]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.5, constant coefficient homogeneous system II. Page 555
Problem number : section 10.5, problem 14
Date solved : Monday, January 27, 2025 at 05:44:20 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=15 y_{1} \left (t \right )-9 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=16 y_{1} \left (t \right )-9 y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 5\\ y_{2} \left (0\right ) = 8 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 28

dsolve([diff(y__1(t),t) = 15*y__1(t)-9*y__2(t), diff(y__2(t),t) = 16*y__1(t)-9*y__2(t), y__1(0) = 5, y__2(0) = 8], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (-12 t +5\right ) \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (-144 t +72\right )}{9} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 31

DSolve[{D[ y1[t],t]==15*y1[t]-9*y2[t],D[ y2[t],t]==16*y1[t]-9*y2[t]},{y1[0]==5,y2[0]==8},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to e^{3 t} (5-12 t) \\ \text {y2}(t)\to -8 e^{3 t} (2 t-1) \\ \end{align*}