Given the second order homogeneous constant coefficient equation y6y=0y6y=0 1) t

Question

Given the second order homogeneous constant coefficient equation y6y=0y6y=0

1) the characteristic polynomial ar2+br+car2+br+c is  .

2) The roots of auxiliary equation are  (enter answers as a comma separated list).

3) A fundamental set of solutions is  (enter answers as a comma separated list).

4) Given the initial conditions y(0)=4y(0)=4 and y(0)=6y(0)=6 find the unique solution to the IVP

y=y=  .

(1 point) Given the second order homogeneous constant coefficient equation y" 6y' -0 1) the characteristic polynomial ar br c is 2) The roots of auxiliary equation are (enter answers as a comma separated list) 3) A fundamental set of solutions is (enter answers as a comma separated list) 4) Given the initial conditions y(0) -4 and y (0) 6 find the unique solution to the IVP

Explanation / Answer

y'' - 6y' = 0

1) The characteristic polynomial is r2 - r

2) The auxillary equation is  r2 - r = 0

=> r(r-1) = 0

=> r = 0 or r = 1

The roots of the auxillary equation are 0,1

3) y = ae0x + be1x

=> y = a + bex

=> A fundamental set of solutions is y = a + bex

4) y(0) = 4 and y'(0) = -6

y = a + bex

=> a + be0 = 0

=> a + b = 0 (1)

y = a + bex

Differentiating,

y' = bex

y'(0) = -6

=> be0 = -6

=> b = -6

Substituting in (1)

=> a - 6 = 0

=> a = 6

=> y = 6 - 6ex

=> y = 6(1-ex)

The unique solution to the IVP is y = 6(1-ex)

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