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Full question: Find the particular solution of the equation $f'(x) = 4x^{-1/2}$ that satisfies the condition $f(1) = 12$.

I have $f'(x) = 4x^{-1/2}$ and am given that $f(1) = 12$.

I took the integral of $4x^{-1/2}$ and got $8x^{1/2}$. I know the integral is correct but when I plug 1 into $8x^{1/2}$, I do not get $12$.

What am I doing wrong?

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up vote 7 down vote accepted

When you integrate an indefinite integral, you need to include a constant of integration to the result.

So, you need to evaluate $f(1) = 12$ given $f(x) = 8x^{1/2} + C$ to determine the required constant C.

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Downvoter: Any reason for the downvote other than personal spite? – amWhy Apr 20 '14 at 16:04
same downvoter on my post, too. – Michael Tong Apr 20 '14 at 16:04
So I would do $8(1)^{1/2}$ + C = 12? Which would be 4. – m.o Apr 20 '14 at 16:10
Yes, exactly. So the particular solution would be $f(x) = 8^{1/2} + 4$. – amWhy Apr 20 '14 at 16:13
Thank you for your help! – m.o Apr 20 '14 at 16:13

You forgot to add your constant! You should get the antiderivative to be $f'(x) = 8x^{1/2} + C = 12$ at $x = 1$, so then solve for $C$.

Indeed, when you anti-differentiate a function, you end up with a whole family of functions each differing by a constant. This is why you need to find the particular solution.

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Thank you for helping as well! I'll try not to forget the constant next time. – m.o Apr 20 '14 at 16:14

If $f'(x)=4 \cdot x^{-\frac{1}{2}}$ then

$f(x)=\text{integration of } f'(x)dx$

which is equal to $8 \cdot x^{\frac{1}{2}}+c \tag{1} $

Here c is some constant.

Now given that $f(1) = 12$ (given above line):

$f(1) = 8\cdot 1^{\frac{1}{2}} + c = 12$

Therefore from above line we can say that $c=4 \tag{2}$

Therefore from equation (1) & (2)

The particular solution of the given function is

$f(x) = 8\cdot x^{\frac{1}{2}} + 4$

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