Differential Equations and Their Solutions

The video tutorial explains how to determine which functions are solutions to the differential equation y'' - y = 0. It evaluates five functions: y = e^x, y = 0, y = sin(x), y = cos(x), and y = 4e^x. The tutorial demonstrates that y = e^x and y = 4e^x are solutions, while y = 0, y = sin(x), and y = cos(x) are not. The process involves finding the second derivative of each function and substituting into the differential equation to check if the result equals zero.

What is the differential equation that the given functions need to satisfy?

• y'' + 2y = 0

• y' - y = 0

• y'' - y = 0

• y'' + y = 0

What is the second derivative of y = e^x?

• e^x

• 2e^x

• 0

• -e^x

Why is y = e^x a solution to the differential equation?

• Because y'' - y = 2e^x

• Because y'' - y = e^x

• Because y'' - y = 0

• Because y'' - y = -e^x

What is the value of the second derivative of y = 0?

• -1

• e^x

• 1

• 0

Why is y = 0 a solution to the differential equation?

• Because y'' - y = 1

• Because y'' - y = 0

• Because y'' - y = e^x

• Because y'' - y = -1

What is the second derivative of y = sin(x)?

• sin(x)

• -cos(x)

• -sin(x)

• cos(x)

Why is y = sin(x) not a solution to the differential equation?

• Because y'' - y = cos(x)

• Because y'' - y = sin(x)

• Because y'' - y = -2sin(x)

• Because y'' - y = 0

What is the second derivative of y = cos(x)?

• -sin(x)

• cos(x)

• -cos(x)

• sin(x)

Why is y = cos(x) not a solution to the differential equation?

• Because y'' - y = 0

• Because y'' - y = sin(x)

• Because y'' - y = 2cos(x)

• Because y'' - y = -cos(x)

Why is y = 4e^x a solution to the differential equation?

• Because y'' - y = 4e^x

• Because y'' - y = 0

• Because y'' - y = 2e^x

• Because y'' - y = -4e^x