9.4.27 problem problem 38

Internal problem ID [991]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 38
Date solved : Monday, January 27, 2025 at 03:22:46 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 74

dsolve([diff(x__1(t),t)=x__1(t)+0*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__2(t),t)=2*x__1(t)+2*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+3*x__2(t)+3*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+4*x__3(t)+4*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_4 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -2 c_4 \,{\mathrm e}^{t}+{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{2 t} c_3 +3 c_4 \,{\mathrm e}^{t} \\ x_{4} \left (t \right ) &= c_1 \,{\mathrm e}^{4 t}-4 c_2 \,{\mathrm e}^{3 t}+6 \,{\mathrm e}^{2 t} c_3 -4 c_4 \,{\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 128

DSolve[{D[ x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],D[ x2[t],t]==2*x1[t]+2*x2[t]+0*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]+3*x2[t]+3*x3[t]+0*x4[t],D[ x4[t],t]==0*x1[t]+0*x2[t]+4*x3[t]+4*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

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