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

Answer:

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:

$y=ce^t\\y'=ce^t\\y'=y$

Step-by-step 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:

$y=ce^t\\y'=\frac{d(ce^t)}{dt} \\y'=ce^t\\y'=y$

Where:

c is any number.

Second Part:

Consider the constant A.

The function will become:

$y=ce^{At}\\y'=\frac{d(ce^{At})}{dt} \\y'=cAe^{At}\\y'=Ay$

Third Part:

Derivative is represented as:

$y=ce^t\\y'=\frac{d(ce^t)}{dt} \\y'=ce^t\\y'=y$

Where:

c is any number.