Result Answer Preview Entered At least one of the answers ab

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

Solution

Ans

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