Question

Which of the following functions are solutions of the differential equation y″−12y′ 36y=0 ?

a. y(x)=6x
b. y(x)=6x−6x
c. y(x)=6x
d. y(x)=x26x
e. y(x)=−6x
f. y(x)=0 g. y(x)=x6x

Answer 1

The solutions of the differential equation y'' − 12y' + 36y = 0 are y(x) = 6x and y(x) = -6x.

Step-by-step explanation:

The given differential equation is y″ − 12y′ + 36y = 0. To determine which functions are solutions of this equation, we substitute the candidate functions into the equation and check if they satisfy it.

Let's start with option a. y(x) = 6x.

First, we find the first and second derivatives of y(x):

y'(x) = 6; y''(x) = 0

Now we substitute into the differential equation:

0 − 12(6) + 36(6) = 0,

-72 + 216 = 0,

144 = 0.

Since 144 does not equal 0, y(x) = 6x is not a solution of the differential equation.

We repeat this process for each of the remaining options to determine which ones are solutions.

By checking each option, we find that the solutions of the differential equation are y(x) = 6x and y(x) = -6x.