Solve a differential equation involving matrices. Let A = \left[ \begin{matrix} -1 & 1 \\ 0 &

ngihlungeqtr

ngihlungeqtr

Answered question

2022-04-20

Solve a differential equation involving matrices.
Let A=-110-1,b=23

Solve the differential equation
x˙=Ax+b;  x(0)=12.
Use formula 

x(t)=etAx(0)+0te(ts)Ab ds =et[1+2t2]+0te(ts)[1ts01][23]ds 
My question is how do we get from the second summand on the RHS on line 1 to the second summand on the RHS of the equation on line 2? That is how does the equation come out?
0te(t-s)Abds=0te-(t-s)1t-s0123ds

Answer & Explanation

Sergio Kidd

Sergio Kidd

Beginner2022-04-21Added 14 answers

Explanation:
e(ts)A=e(ts)etse(ts)A=e(ts)e(ts)(A+I)=e(ts)e[0ts00]=e(ts)(I+[0ts00])=e(ts)[1ts01]where I have used formula eA=I+A, which is valid if A2=0.

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