Question

What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple of itself? Write each answer in the form of a first-order differential equation with a solution.

Answer 1

First Part:

exponential function
`e^t` is the one whose first order derivative is the function itself.

Second Part:

`y=ce^(At)\\y'=Ay`

Third Part:

`y=ce^t\\y'=ce^t\\y'=y`

Explanation:

First Part:

In calculus exponential function
`e^t` is the one whose first order derivative is the function itself.

Where:

t is independent variable.

Derivative is represented as:

`y=ce^t\\y'=(d(ce^t))/(dt) \\y'=ce^t\\y'=y`

Where:

c is any number.

Second Part:

Consider the constant A.

The function will become:

`y=ce^(At)\\y'=(d(ce^(At)))/(dt) \\y'=cAe^(At)\\y'=Ay`

Third Part:

Derivative is represented as:

`y=ce^t\\y'=(d(ce^t))/(dt) \\y'=ce^t\\y'=y`

Where:

c is any number.