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By eliminating constants a and b, form a differential equation from the function y = ae⁴x be^-x?

User Hemaulo
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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''.

User Arpit Kulsreshtha
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