12.22.12 problem section 10.5, problem 12

Internal problem ID [2265]
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 12
Date solved : Monday, January 27, 2025 at 05:44:18 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=6 y_{1} \left (t \right )-5 y_{2} \left (t \right )+3 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=2 y_{1} \left (t \right )-y_{2} \left (t \right )+3 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=2 y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 85

dsolve([diff(y__1(t),t)=6*y__1(t)-5*y__2(t)+3*y__3(t),diff(y__2(t),t)=2*y__1(t)-1*y__2(t)+3*y__3(t),diff(y__3(t),t)=2*y__1(t)+1*y__2(t)+1*y__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 127

DSolve[{D[ y1[t],t]==6*y1[t]-5*y2[t]+3*y3[t],D[ y2[t],t]==2*y1[t]-1*y2[t]+3*y3[t],D[ y3[t],t]==2*y1[t]+1*y2[t]+1*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{2} e^{-2 t} \left (e^{6 t} (c_1 (4 t+2)-c_2 (4 t+1)+c_3)+c_2-c_3\right ) \\ \text {y2}(t)\to \frac {1}{2} e^{-2 t} \left (e^{6 t} (4 (c_1-c_2) t+c_2+c_3)+c_2-c_3\right ) \\ \text {y3}(t)\to \frac {1}{2} e^{-2 t} \left (e^{6 t} (4 (c_1-c_2) t+c_2+c_3)-c_2+c_3\right ) \\ \end{align*}