Result Answer Preview Entered At least one of the answers above is NOT corredt 2

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Result Answer Preview Entered At least one of the answers above is NOT corredt 2 of the questions remain unanswered. (1 point) Math 216 Homework F16 216_WebWork 5, Problem Consider the homogeneous second-order linear differential equation Which of the following pairs gives two solutions to this equation? Then for these solutions find a particular solution of the form that satisfies the initial conditions (o)-7, (0o. Note: You can eam partiaí credit on this probilem Preview My AnewersSubmit Answers Home 5 6 8 9

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Let ,

y" + 8 y' +15y = 0

The auxillary equation is

r2 + 8r + 15 = 0

r2 +5r + 3r + 15 = 0

(r+5) (r+ 3) = 0

(r+5) =0 , (r+ 3) =0

whose roots are r = -3 , r = -5

Therefore the general solution of the given differential equation is

y = c1 e-3x + c2 e-5x

Therefore the answer B is the solution

i. e y1  = e-3x   , y2 = e-5x

Let ,

y = c1 y1 + c2 y2  with initial condition y(0) = -7 , y' (0) = 0

Let , y(x) = c1 e-3x + c2 e-5x -------------------------------- 1

the derivative of the solution is

y' (x) = -3c1e-3x  - 5 c2 e-5x   -------------------------------------------2

substituting x =0 in 1 and 2 with initial condition y(0) = -7 , y'(0) = 0

The system of equation is

c1 + c2  = -7

-3c1 - 5 c2 = 0

solving system we get

c1 = -35/2 and c2 = 21/2

Therefore the solution is

y(x) = (-35/2) e-3x + (21/2) e-5x

i.e y(x) = (-35/2) y1  + (21/2) y2

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