Question

Verify that the indicated family of functions is a solution of the given differential equation. assume an appropriate interval i of definition for each solution. d2y dx2 − 8 dy dx + 16y = 0; y = c1e4x + c2xe4x

Answer 1

We verified that the family of functions y = c1e^(4x) + c2xe^(4x) is a solution to the differential equation d2y/dx2 - 8dy/dx + 16y = 0 by calculating the first and second derivatives, substituting them into the differential equation, and showing that it is satisfied.

Step-by-step explanation:

We have been given the differential equation d2y/dx2 - 8dy/dx + 16y = 0 and a family of functions y = c1e4x + c2xe4x. We need to verify that the given family of functions is a solution to the differential equation. Let’s calculate the first and second derivatives of the proposed solution.

First derivative (dy/dx):

1. Differentiating y = c1e4x + c2xe4x with respect to x gives dy/dx = 4c1e4x + c2e4x + 4c2xe4x.
2. Simplify to get dy/dx = (4c1 + c2)e4x + 4c2xe4x.
   

Second derivative (d2y/dx2):

1. Now, differentiate dy/dx with respect to x, d2y/dx2 = 16c1e4x + 4c2e4x + 4c2e4x + 16c2xe4x.
2. Simplify to get d2y/dx2 = (16c1 + 8c2)e4x + 16c2xe4x.

Substitute the first and second derivatives back into the differential equation to confirm:

• (16c1 + 8c2)e4x + 16c2xe4x - 8((4c1 + c2)e4x + 4c2xe4x) + 16(c1e4x + c2xe4x) = 0. Simplifying the equation, all terms cancel out, confirming that y = c1e4x + c2xe4x is indeed a solution of the differential equation.

Answer 2

Given:
`y = c_(1)e^(4x)+c_(2)xe^(4x)`

`{y}'' - 8 {y}' + 16y = 0` ..... (1)

We can rewrite it as:

`y = \left (c_(1)+c_(2)x  \right )e^(4x)`

Find the derivatives.

`{y}' = \frac{\mathrm{d} y}{\mathrm{d} x}`

`= \left (c_(1)+c_(2)x  \right ) 4e^(4x) + 4c_(2)e^(4x)4`

`= \left (4c_(1) + c_(2) + 4 c_(2)x  \right ) e^(4x)`

`{y}'' =  \left (4c_(1) + c_(2) + 4 c_(2)x  \right ) 4e^(4x) +  4c_(2)e^(4x)`

`= \left (16c_(1) + 8c_(2) + 16 c_(2)x  \right )e^(4x)`

Substitute the values of
`y, {y}' and {y}''` in equation (1)

`{y}'' - 8{y}' + 16y =  \left (16c_(1) + 8c_(2) + 16 c_(2)x  \right )e^(4x) - 8 \left (4c_(1) + c_(2) + 4 c_(2)x  \right ) e^(4x) + 16 \left (c_(1)+c_(2) x  \right )e^(4x)`

`= \left \{ 16c_(1) + 8c_(2) + 16c_(2)x - 32c_(1) - 8c_(2) -32c_(2)x + 16c_(1) + 16 c_(2)x \right \}e^(4x)`

`= 0`

Hence Proved.