Question

In Problems 1–10 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.

8. y'''+4y''+3y'=x² cosx^_3x

Short answer

The required form is D(D+1)(D+3)y=x²cosx^_3x.

Step-by-step solution

Definition of differential equation.

An equation containing the derivatives of one or more unknown functions (or dependent variables) with respect to one or more independent variables is said to be a differential equation (DE).

Solve the differential equation.

The given differential equation is

y'''+4y''+3y=x²cosx^_3x

Write it in the form L(y) =g(x) using the differential linear operators as follows:

D³y+4D²y+3dy =x²cosx^_3x

(D³+4D²+3D)y = x²cosx^_3x

Now, factorize (D³+4D²+3D) as follows:

(D³+4D²+3D) = D(D²+4D+3)

= D(D+1)(D+3)

So, the differential equation is written as:

D(D+1)(D+3)y=x²cosx ^_3x

Where L=D(D+1)(D+3) and g(x) =x²cosx ^_3x.