9.6.30 problem problem 30

Internal problem ID [1037]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 30
Date solved : Monday, January 27, 2025 at 03:23:01 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{2} \left (t \right )-5 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-13 x_{2} \left (t \right )+22 x_{3} \left (t \right )-12 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-27 x_{2} \left (t \right )+45 x_{3} \left (t \right )-25 x_{4} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 88

dsolve([diff(x__1(t),t)=2*x__1(t)+1*x__2(t)-2*x__3(t)+1*x__4(t),diff(x__2(t),t)=0*x__1(t)+3*x__2(t)-5*x__3(t)+3*x__4(t),diff(x__3(t),t)=0*x__1(t)-13*x__2(t)+22*x__3(t)-12*x__4(t),diff(x__4(t),t)=0*x__1(t)-27*x__2(t)+45*x__3(t)-25*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\left (-c_2 t +3 c_1 \right ) {\mathrm e}^{2 t}}{3} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (c_4 t +c_3 \right ) \\ x_{3} \left (t \right ) &= \left (-{\mathrm e}^{-3 t} \left (c_4 t +c_3 -c_4 \right )+c_2 \right ) {\mathrm e}^{2 t} \\ x_{4} \left (t \right ) &= -3 c_3 \,{\mathrm e}^{-t}-3 c_4 \,{\mathrm e}^{-t} t +2 \,{\mathrm e}^{-t} c_4 +\frac {5 c_2 \,{\mathrm e}^{2 t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 161

DSolve[{D[ x1[t],t]==2*x1[t]+1*x2[t]-2*x3[t]+1*x4[t],D[ x2[t],t]==0*x1[t]+3*x2[t]-5*x3[t]+3*x4[t],D[ x3[t],t]==0*x1[t]-13*x2[t]+22*x3[t]-12*x4[t],D[ x4[t],t]==0*x1[t]-27*x2[t]+45*x3[t]-25*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} ((c_2-2 c_3+c_4) t+c_1) \\ \text {x2}(t)\to e^{-t} (4 c_2 t-5 c_3 t+3 c_4 t+c_2) \\ \text {x3}(t)\to e^{-t} \left (c_2 \left (-4 t-3 e^{3 t}+3\right )+c_3 \left (5 t+6 e^{3 t}-5\right )-3 c_4 \left (t+e^{3 t}-1\right )\right ) \\ \text {x4}(t)\to e^{-t} \left (c_2 \left (-12 t-5 e^{3 t}+5\right )+5 c_3 \left (3 t+2 e^{3 t}-2\right )-c_4 \left (9 t+5 e^{3 t}-6\right )\right ) \\ \end{align*}