Question

In Problems 13 and 14 determine by inspection at least one solution of the given differential equation.

y' = y(y - 3)

Short answer

The solutions for the preceding differential equation are y = 0 or y = 3.

Step-by-step solution

Define second derivative of a function

The derivative of the derivative of a function f is known as the second derivative, or second order derivative, in calculus.

So, the second derivative, or the rate of change of speed with respect to time, can be used to determine the variation in speed of the car (the second derivative of distance travelled with respect to the time).

Determine the second derivative of the function

For the Find the solution such that the left-hand limit is equal to the right-hand limit of a function. The solutions for the preceding differential equation are y = 0 or y=3 .

Let check this:

As the constant function,y = 0then, the left-hand limit isy’=0, and the right-hand limit is,y(y-3)=0(0-3)

Therefore,y' = y(y - 3).

Also, as the constant function,y = 3then the left-hand limit isy' = 0, and the right-hand limit is,y(y - 3) = 3(3 - 3) = 0.

Therefore,y' = y(y - 3).

Hence, y = 0 and y = 3.