Solution 1 Algebraic properties of solutions of linear systems choice of constants c1 c2 c1 For example consider the system of equations dx 0 dt 4
Solution Algebraic properties of solutions of linear systems choice of constants c c c For example consider the system
Solution Algebraic properties of solutions of linear systems choice of constants c c c
solutions of linear systems choice of constants c c c For example consider the system of equations dx dt
Solution Algebraic properties of solutions of linear systems choice of constants c
c c For example consider the system of equations dx dt
Solution Algebraic properties of solutions of linear systems choice
Solution Algebraic properties of
(Solution) 1 Algebraic properties of solutions of linear systems choice of constants c1,c2, ,c1. For example, consider the system of equations dx ( 0 dt = -4...

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Please complete page 271 #1,2,5,6,9,10,11,13,14,15 (from Braun4e-287-289)I would like thorough explanations/work shown for all the problems solved. If you have any questions, let me know.3.1 Algebraic properties of solutions of linear systems choice of constants c 1 ,c 2 , ••• ,c 1 . For example, consider the system of equa- tions dx ( 0 dt = -4 b )x, x = ( ~~). (6) This system of equations was derived from the second-order scalar equa- tion (d 2 y I dt 2 )+4y =0 by setting x 1 = y and x 2 = dy I dt. Since y 1 (t)=cos2t and yit) = sin2t are two solutions of the scalar equation, we know that x(t) = ( x 1 (t)) = c 1 ( cos2t) + c 2 ( sin2t) x 2 (t) -2sin2t 2cos2t ( c 1 cos2t+c 2 sin2t) = - 2c 1 sin2t + 2c 2 cos2t is a solution of (6) for any choice of constants c 1 and c 2 • The next step in our gameplan is to show that every solution of (4) can be expressed as a linear combination of finitely many solutions. Equiv- alently, we seek to determine how many solutions we must find before we can generate all the solutions of (4). There is a branch of mathematics known as linear algebra, which addresses itself to exactly this question, and it is to this area that we now turn our attention. EXERCISES In each of Exercises 1-3 convert the given differential equation for the sin- gle variable y into a system of first-order equations. 1. d3y+(dy)2=0 2. d3y+cosy=e' 3. d4y+d2y=l dt 3 dt dt 3 dt 4 dt 2 4. Convert the pair of second-order equations d 2 y dz d 2 dy 3 2 0 ~ + 3- + 2z = 0 dt2 + dt + y = ' dt2 dt into a system of 4 first-order equations for the variables xl=y, 5. (a) Lety(t) be a solution of the equationy"+y'+y=O. Show that x(t)=(y(t)) y'(t) is a solution of the system of equations x=( -~ -Dx. 271