12.21.15 problem section 10.4, problem 15

Internal problem ID [2253]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 15
Date solved : Monday, January 27, 2025 at 05:44:09 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=3 y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=3 y_{1} \left (t \right )+5 y_{2} \left (t \right )+y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-6 y_{1} \left (t \right )+2 y_{2} \left (t \right )+4 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)+1*y__2(t)-1*y__3(t),diff(y__2(t),t)=3*y__1(t)+5*y__2(t)+1*y__3(t),diff(y__3(t),t)=-6*y__1(t)+2*y__2(t)+4*y__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 93

DSolve[{D[ y1[t],t]==3*y1[t]+1*y2[t]-1*y3[t],D[ y2[t],t]==3*y1[t]+5*y2[t]+1*y3[t],D[ y1[t],t]==-6*y1[t]+2*y2[t]+4*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{625} \left (-\left (c_1 \left (e^{6 t}-500\right )\right )-c_2 \left (e^{6 t}+125\right )\right ) \\ \text {y2}(t)\to \frac {1}{625} \left (c_2 \left (125-4 e^{6 t}\right )-4 c_1 \left (e^{6 t}+125\right )\right ) \\ \text {y3}(t)\to \frac {1}{625} \left (-\left (c_1 \left (e^{6 t}-1000\right )\right )-c_2 \left (e^{6 t}+250\right )\right ) \\ \end{align*}