10.18.5 problem 5

Internal problem ID [1432]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:57:14 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )+\frac {1}{t^{3}}\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )-4 x_{2} \left (t \right )-\frac {1}{t^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=4*x__1(t)-2*x__2(t)+1/(t^3),diff(x__2(t),t)=8*x__1(t)-4*x__2(t)-1/(t^2)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {1}{2 t^{2}}+\frac {2}{t}-2 \ln \left (t \right )+c_1 t +c_2 \\ x_{2} \left (t \right ) &= 2 c_1 t -4 \ln \left (t \right )-\frac {c_1}{2}+2 c_2 +\frac {5}{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 61

DSolve[{D[ x1[t],t]==4*x1[t]-2*x2[t]+1/(t^3),D[ x2[t],t]==8*x1[t]-4*x2[t]-1/(t^2)},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -\frac {1}{2 t^2}+\frac {2}{t}-2 \log (t)+4 c_1 t-2 c_2 t-2+c_1 \\ \text {x2}(t)\to \frac {5}{t}-4 \log (t)+8 c_1 t-4 c_2 t-4+c_2 \\ \end{align*}