12.22.31 problem section 10.5, problem 31

Internal problem ID [2284]
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 31
Date solved : Monday, January 27, 2025 at 05:44:34 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-3 y_{1} \left (t \right )-3 y_{2} \left (t \right )+4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=4 y_{1} \left (t \right )+5 y_{2} \left (t \right )-8 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=2 y_{1} \left (t \right )+3 y_{2} \left (t \right )-5 y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 52

dsolve([diff(y__1(t),t)=-3*y__1(t)-3*y__2(t)+4*y__3(t),diff(y__2(t),t)=4*y__1(t)+5*y__2(t)-8*y__3(t),diff(y__3(t),t)=2*y__1(t)+3*y__2(t)-5*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_3 t +c_2 \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (-2 c_3 t +c_1 -2 c_2 \right ) \\ y_{3} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-4 c_3 t +3 c_1 -4 c_2 +c_3 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 83

DSolve[{D[ y1[t],t]==-3*y1[t]-3*y2[t]+4*y3[t],D[ y2[t],t]==4*y1[t]+5*y2[t]-8*y3[t],D[ y3[t],t]==2*y1[t]+3*y2[t]-5*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to e^{-t} (-2 c_1 t-3 c_2 t+4 c_3 t+c_1) \\ \text {y2}(t)\to e^{-t} ((4 c_1+6 c_2-8 c_3) t+c_2) \\ \text {y3}(t)\to e^{-t} ((2 c_1+3 c_2-4 c_3) t+c_3) \\ \end{align*}