Question

Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. (Use D for the differential operator.) y(4) + 8y' = 6

Answer 1

Explanation:

Given the differential equation

y^(4) + 8y' = 6

We want to write this in the form

L(y) = g(x)

Where L is a linear differential operator with constant coefficient.

A linear differential operator of the nth order is a linear combination of derivative operators up to n.

L = D^n + a_1D^(n-1) + a_2D^(n-2) + ... + a_n,

defined by

Ly = y^n + a_1y^(n-1) + a_2y^(n-2) + ... + a_(n-1)y' + a_ny

Where a_i are continuous functions of x.

Now, we have

y^(4) + 8y' = 6

Let d/dx = D

Then

D^4 y + 8Dy = 6

D(D³ + 8)y = 6

Consider

D(D³ + 8)y = 0

The auxiliary equation is

m(m³ + 8) = 0

m = 0

Or

m³ + 8 = 0

=> m³ = -8

=> m = -2

The complimentary solution is

y = C1 + (C2 + C3x + C4x²)e^(-2x)

The particular integral is

y_p = Ax

y' = A

y'' = y''' = y^(4) = 0

Using these

0 + 8A = 6

A = 6/8 = 3/4

So

y = C1 + (C2 + C3x + C4x²)e^(-2x) + 3/4

Answer 2

The differential equation will be like the one shown below

Explanation:

Data:

Let the equation be given as:

y(4) + 8y' = 6

The equation will be expressed linearly as follows:

y(4) + 8
`(dy)/(dx) = 6`

`8(dy)/(dx)+ 4y = 6`

This is the linear form of the differential equation.