I've just started learning about integrals... so what does it mean by: "Solve the differential equation y' = 4x^(-1/2) at point (1, 12)" ?

Question

abb0t · Answer

What it means is that you have: $\sf \frac{dy}{dx}=4x^{-\frac{1}{2}} $
this is called a seperable differential equation.

abb0t · Answer

(1,12) is your initial conditions.

jigglypuff314 · Answer

so I find the anti derivative and plug in (1,12) to find C ?

abb0t · Answer

I mean, you can think of it as "cross-multiplying" but you shouldn't be thinking like that now at differential equations level. Lol

$$\sf dy=(4x^{-\frac{1}{2}})~dx$$

abb0t · Answer

You separated them into their respective variable. Now, how do you solve for any antiderivative? You integrate BOTH SIDES

jigglypuff314 · Answer

I can get that   y = 8x^(1/2) + C
I was just confused at the terminology of  What is the question asking for?

abb0t · Answer

$$\sf \int dy = \int (4x^{-\frac{1}{2}})dy$$

abb0t · Answer

Sorry, the second variable should be $dx$ not $\color{red}{dy}$

ganeshie8 · Answer

more generally the question is begging u to find a function  $y$ with below properties:

1) the function $y$ has a slope $4x^{-1/2}$ 
2) function passes thru the point $ (1,2)$

jigglypuff314 · Answer

so would the equation be   y = 8x^(1/2) + 4  ?

abb0t · Answer

That is a relatively simple integral which you should be able to solve. Then, plug in the initial conditions
y = 12, x = 1

and solve for C.