Question

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

Answer 1

1.

$$\dfrac{dy}{dx}=y$$

$$\dfrac{dy}{y}=dx$$

Integrate

$$\int\dfrac{dy}{y}=\int dx$$

$$\ln|y|=x+\ln C$$

$$y=Ce^x$$

2.

$$\dfrac{dy}{dx}=ky$$

$$\dfrac{dy}{y}=kdx$$

Integrate

$$\int\dfrac{dy}{y}=\int kdx$$

$$\ln|y|=kx+\ln C$$

$$y=Ce^{kx}$$