Given f(x) = 2x , f(2) = 1 , f(1) = 3 , find the particular solution to the differential equation.

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We are given f''(x)=2x, f'(2)=1, and f(1)=3 and we are asked to find the particular solution to the differential equation.

Since we know the second derivative, we can use the indefinite integral (or antidifferentiation) to find that `f'(x)=x^2+C` . C is called the constant of integration; we need it because a family of functions has the same derivative where the family of functions differ only by a constant.

Since we are given f'(2)=1, we can solve for C. Substituting 2 for x and...

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