Question

What function do you know from calculus is such that its first derivative is itself? (Do not use the function f(x) = 0.)

f(x) = ___

The above function is a solution of which of the following differential equations?

A. y prime = y
B. y prime = y^2
C. y prime = 1
D. y prime = e^y
E. y prime = 2y

What function do you know from calculus is such that its first derivative is a constant multiple k of itself? (Do not use the function
f(x) = 0.)

f(x) = ___

The above function is a solution of which of the following differential equations?

A. y prime = k
B. y prime = y + k
C. y prime = e^ky
D. y prime = ky
E. y prime = y^k

Answer 1

The function for which first derivative is itself is $e^x$. This function is the solution of the equation $y'=y$, since it is given that the first derivative is itself. Hence option (A) is the correct option.

The function for which first derivative is constant multiple of $k$ of itself is $e^{kx}$ . This function is the solution of the equation $y'=ky$, since it is given that the first derivative is a constant multiple of $k$ to itself. Hence option (D) is the correct option.