Question

Verify whether these functions are solutions to these differential equations.

a. y = e^(-x) + Ce^(-2x), (dy/dx) + 2y = e^(-x)
b. y = 3x^(-2) + Cx^(-4), (4x²y - 6)dx + x³dy = 0
c. e^y - e^x = C, (dy/dx) = e^(x-y)

Answer 1

To determine if the functions are solutions to their differential equations, each function must be differentiated and then substituted back into the equation to verify if the equation is true.

Step-by-step explanation:

To verify if the given functions are solutions to their respective differential equations, we must substitute each function into the equation and check if the equation holds.

1. For function a: y = e^(-x) + Ce^(-2x), we need to find the derivative (dy/dx) and substitute into the differential equation (dy/dx) + 2y = e^(-x) to check for consistency.
2. For function b: y = 3x^(-2) + Cx^(-4), we need to differentiate y with respect to x and check the given differential equation (4x^2y - 6)dx + x^3dy = 0 to ensure it is satisfied.
3. For function c: e^y - e^x = C, we need to differentiate both sides with respect to x and see if it equals (dy/dx) = e^(x-y).

Through this process, we can determine which function satisfies its differential equation and which does not.