12.22.25 problem section 10.5, problem 25

Internal problem ID [2278]
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 25
Date solved : Monday, January 27, 2025 at 05:44:30 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=y_{1} \left (t \right )+10 y_{2} \left (t \right )-12 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=2 y_{1} \left (t \right )+2 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 )+6 y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 85

dsolve([diff(y__1(t),t)=1*y__1(t)+10*y__2(t)-12*y__3(t),diff(y__2(t),t)=2*y__1(t)+2*y__2(t)+3*y__3(t),diff(y__3(t),t)=2*y__1(t)-1*y__2(t)+6*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (6 c_3 \,t^{2}+6 c_2 t +6 c_3 t +6 c_1 +3 c_2 +4 c_3 \right )}{6} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (18 c_3 \,t^{2}+18 c_2 t +18 c_3 t +18 c_1 +9 c_2 +10 c_3 \right )}{18} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 118

DSolve[{D[ y1[t],t]==1*y1[t]+10*y2[t]-12*y3[t],D[ y2[t],t]==2*y1[t]+2*y2[t]+3*y3[t],D[ y3[t],t]==2*y1[t]-1*y2[t]+6*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to -e^{3 t} (c_1 (2 t-1)+c_2 t (9 t-10)+3 c_3 (4-3 t) t) \\ \text {y2}(t)\to e^{3 t} \left (9 (c_2-c_3) t^2+(2 c_1-c_2+3 c_3) t+c_2\right ) \\ \text {y3}(t)\to e^{3 t} \left (9 (c_2-c_3) t^2+(2 c_1-c_2+3 c_3) t+c_3\right ) \\ \end{align*}