76.29.8 problem 8

Internal problem ID [17884]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.7 (Defective Matrices). Problems at page 444
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 11:09:42 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )-2 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-\frac {3 x_{2} \left (t \right )}{2}-x_{3} \left (t \right )+\frac {7 x_{4} \left (t \right )}{2}\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{1} \left (t \right )+\frac {x_{2} \left (t \right )}{2}-\frac {3 x_{4} \left (t \right )}{2}\\ \frac {d}{d t}x_{4} \left (t \right )&=-2 x_{1} \left (t \right )+\frac {3 x_{2} \left (t \right )}{2}+3 x_{3} \left (t \right )-\frac {7 x_{4} \left (t \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.131 (sec). Leaf size: 110

dsolve([diff(x__1(t),t)=1*x__1(t)-1*x__2(t)-2*x__3(t)+3*x__4(t),diff(x__2(t),t)=2*x__1(t)-3/2*x__2(t)-1*x__3(t)+7/2*x__4(t),diff(x__3(t),t)=-1*x__1(t)+1/2*x__2(t)-0*x__3(t)-3/2*x__4(t),diff(x__4(t),t)=-2*x__1(t)+3/2*x__2(t)+3*x__3(t)-7/2*x__4(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 206

DSolve[{D[x1[t],t]==1*x1[t]-1*x2[t]-2*x3[t]+3*x4[t],D[x2[t],t]==2*x1[t]-3/2*x2[t]-1*x3[t]+7/2*x4[t],D[x3[t],t]==-1*x1[t]+1/2*x2[t]-0*x3[t]-3/2*x4[t],D[x4[t],t]==-2*x1[t]+3/2*x2[t]+3*x3[t]-7/2*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (c_1 \left (-t^2+2 t+1\right )+t (c_2 (t-1)+2 c_3 (t-1)-c_4 (t-3))\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-3 (c_1-c_2-2 c_3+c_4) t^2+(4 c_1-c_2-2 c_3+7 c_4) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left ((c_1-c_2-2 c_3+c_4) t^2+(-2 c_1+c_2+2 c_3-3 c_4) t+2 c_3\right ) \\ \text {x4}(t)\to \frac {1}{2} e^{-t} \left ((c_1-c_2-2 c_3+c_4) t^2+(-4 c_1+3 c_2+6 c_3-5 c_4) t+2 c_4\right ) \\ \end{align*}