Final answer:
To form the differential equation from the given function y = ae^4x + be^-x, you differentiate the function two times and solve the resulting system of equations to eliminate the constants a and b.
Step-by-step explanation:
To form a differential equation from the function y = ae4x + be-x, where a and b are constants, we need to differentiate the function enough times to eliminate these constants. Let's differentiate the function with respect to x twice, as there are two constants to eliminate.
First Derivative:
y' = 4ae4x - be-x
Second Derivative:
y'' = 16ae4x + be-x
Now we have a system of three equations:
- y = ae4x + be-x
- y' = 4ae4x - be-x
- y'' = 16ae4x + be-x
By solving this system, we can eliminate constants a and b and obtain a differential equation that only involves y, y', and y''.