Question

Solve the differential equation y prime =4x sqrt( 1-y^2).

Answer 1

Try separating the variables then integrating.

Answer 2

Looks separable to me

\[\large \frac{dy}{dx} = 4x\sqrt{1 - y^2}\]

\[\large \frac{1}{\sqrt{1 - y^2}}dy = 4xdx\]

Should be good from there!

Answer 3

Hint* Think of inverse trig functions whilst integrating

Answer 4

would the 4x side be just 4?
or would it be 2x^2 ?

Answer 5

You tell me

\[\large \int 4xdx = 4\int xdx = ?\]

Answer 6

arcsine(y) = 2x^2 ?

Answer 7

Perfect....but forgot the +c of course :D

\[\large sin^{-1}(y) = 2x^2 + c\]

So now to solve for 'y' we...?

Answer 8

how do i "get rid" of the sin ^ (-1) ?

Answer 9

Hint* \(\large sin(sin^{-1}(x)) = x\)

Well..really big hint lol

Answer 10

y = sin(2x^2) + c

Answer 11

Uhh, just throw the constant inside of the parenthesis and you're good

Answer 12

Explain why the initial value problem with y(0) = 4 does not have a solution

Answer 13

Is that a second part to this problem?

Answer 14

yes

Answer 15

Okay so think about it...we have

\[\large y = sin(2x^2 + c)\]

If we have y(0) = 4 that means

\[\large y(0) = sin(2(0)^2 + c)\]

\[\large 4 = sin(0 + c)\]
\[\large 4 = sin(c)\]

Tell me a number to plug in for 'c' there that will result in an answer of 4 :)