A. By inspection, find a particular solution of y'' + 2y = 6. yp (x) =

(b) By inspection, find a particular solution of y'' + 2y = - 12x. yp (x) =

(c) Find a particular solution of y'' + 2y = - 12x + 6. yp (x) =

(d) Find a particular solution of y'' + 2y = 24x + 3. yp (x) =

(a) The particular solution, y_p is 3

(b) y_p is - 6x

(c) y_p is - 6x + 3

(d) y_p is 12x + 3

Step-by-step explanation:

To find a particular solution to a differential equation by inspection - is to assume a trial function that looks like the nonhomogeneous part of the differential equation.

(a) Given y'' + 2y = 6.

Because the nonhomogeneus part of the differential equation, 6 is a constant, our trial function will be a constant too.

Let A be our trial function:

We need our trial differential equation y''_p + 2y_p = 6

Now, we differentiate y_p = A twice, to obtain y'_p and y''_p that will be substituted into the differential equation.

y'_p = 0

y''_p = 0

Substitution into the trial differential equation, we have.

0 + 2A = 6

A = 6/2 = 3

Therefore, the particular solution, y_p = A is 3

(b) y'' + 2y = - 12x

Let y_p = Ax + B

y'_p = A

y''_p = 0

0 + 2 (Ax + B) = - 12x

2Ax + 2B = - 12x

By inspection,

2B = 0 = > B = 0

2A = - 12 = > A = - 12/2 = - 6

The particular solution y_p = Ax + B

is - 6x

(c) y'' + 2y = - 12x + 6

Let y_p = Ax + B

y'_p = A

y''_p = 0

0 + 2 (Ax + B) = - 12x + 6

2Ax + 2B = - 12x + 6

By inspection,

2B = 6 = > B = 6/2 = 3

2A = - 12 = > A = - 12/2 = - 6

The particular solution y_p = Ax + B

is - 6x + 3

(d) Find a particular solution of y'' + 2y = 24x + 3

Let y_p = Ax + B

y'_p = A

y''_p = 0

0 + 2 (Ax + B) = 24x + 6

2Ax + 2B = 24x + 6

By inspection,

2B = 6 = > B = 6/2 = 3

2A = 24 = > A = 24/2 = 12

The particular solution y_p = Ax + B

is 12x + 3