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Find the general form of f if f′(x)=4f(x).
f(x)= ...

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Final answer:

The general form of the function f, given that f'(x) = 4f(x), is f(x) = Ce^4x, where C is an arbitrary constant. This form is verified by differentiating to show that it satisfies the original differential equation.

Step-by-step explanation:

The student is tasked with finding the general form of a function f when given the differential equation f'(x) = 4f(x). This is a first-order linear homogeneous differential equation. The solution to this type of equation can be expressed as an exponential function.

To solve this equation, notice that the derivative of f is proportional to the function itself, which is characteristic of exponential growth or decay. The general form of the solution is f(x) = Ce4x, where C is an arbitrary constant. This constant could be determined if an initial condition was given.

We can verify this solution by differentiating f(x) = Ce4x and confirming that indeed f'(x) = 4Ce4x, which satisfies the original differential equation since 4f(x) equals 4Ce4x.

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