Differential Equations and Derivatives

This video tutorial covers solutions to differential equations, demonstrating how to verify if a given function satisfies a differential equation. It includes three examples: verifying a polynomial function, a trigonometric function, and an exponential function as solutions to their respective differential equations. The tutorial explains the use of the power rule, chain rule, and product rule in finding derivatives and performing substitutions to confirm solutions.

What is a solution to a differential equation?

• A function that only satisfies algebraic equations

• Any random function

• A function that satisfies the differential equation on the given interval

• A function that does not satisfy the differential equation

What is the first step in verifying that y = (1/16)x^4 is a solution to the differential equation dy/dx = x * y^(1/2)?

• Find the derivative of y with respect to x

• Solve the equation for x

• Substitute y into the equation directly

• Find the second derivative of y

What is the derivative of y = (1/16)x^4 with respect to x?

• 1/16 x^4

• 1/4 x^3

• x^2

• 4 x^3

In verifying y = 4cos(2x) + 6sin(2x) as a solution, what is the first derivative y'?

• -4cos(2x) + 6sin(2x)

• -8sin(2x) + 12cos(2x)

• 8cos(2x) - 12sin(2x)

• 4sin(2x) + 6cos(2x)

What is the second derivative y'' for y = 4cos(2x) + 6sin(2x)?

• 16cos(2x) + 24sin(2x)

• 8sin(2x) - 12cos(2x)

• -8cos(2x) + 12sin(2x)

• -16cos(2x) - 24sin(2x)

What is the purpose of finding y'' in the verification of y = 4cos(2x) + 6sin(2x)?

• To find the maximum value of y

• To determine the period of the function

• To check if y'' equals zero

• To verify y'' + 4y equals zero

What rule is applied to find the derivative of e^(-x) in y = e^(-x) + 2xe^(-x)?

• Power rule

• Chain rule

• Quotient rule

• Product rule

What is the first derivative dy/dx for y = e^(-x) + 2xe^(-x)?

• e^(-x) + 2xe^(-x)

• e^(-x) - 2xe^(-x)

• -e^(-x) + 2xe^(-x)

• -e^(-x) - 2xe^(-x)

What is the second derivative y'' for y = e^(-x) + 2xe^(-x)?

• 3e^(-x) + 2xe^(-x)

• 3e^(-x) - 2xe^(-x)

• -3e^(-x) + 2xe^(-x)

• -3e^(-x) - 2xe^(-x)

What is the final step in verifying y = e^(-x) + 2xe^(-x) as a solution to the differential equation?

• Determine the limit of y as x approaches infinity

• Find the integral of y

• Verify that the sum of terms equals zero

• Check if y'' equals zero