Question

Find a homogeneous linear differential equation with constant coefficients whose general solution is given by

y=c1e−xcosx+c2e−xsinx.

Answer 1

`y`

Explanation:

Given

`y=c_1e^(-x)cosx+c_2e^(-x)sinx`

Required

Determine a homogeneous linear differential equation

Rewrite the expression as:

`y=c_1e^(\alpha x)cos(\beta x)+c_2e^(\alpha x)sin(\beta x)`

Where

`\alpha = -1` and
`\beta = 1`

For a homogeneous linear differential equation, the repeated value m is given as:

`m = \alpha \± \beta i`

Substitute values for
`\alpha` and
`\beta`

`m = -1 \± 1*i`

`m = -1 \± i`

Add 1 to both sides

`m +1= 1 -1 \± i`

`m +1= \± i`

Square both sides

`(m +1)^2= (\± i)^2`

`m^2 + m + m + 1 = i^2`

`m^2 + 2m + 1 = i^2`

In complex numbers:

`i^2 = -1`

So, the expression becomes:

`m^2 + 2m + 1 = -1`

Add 1 to both sides

`m^2 + 2m + 1 +1= -1+1`

`m^2 + 2m + 2= 0`

This corresponds to the homogeneous linear differential equation

`y`