The question asks to find y given y(x) is a differentiable function satisfying: d y...

prirodnogbk

prirodnogbk

Answered

2022-07-07

The question asks to find y given y(x) is a differentiable function satisfying:
d y d x =- 2 x y 4 , y(0)= 1 3 and y ( x ) > 0 , and explain why it is unique.
I assume the steps are to integrate d y d x to get a function for y(x), then use y ( 0 ) = 1 / 3 to find the constant and then set y ( x ) > 0 to find y but i keep getting stuck.
A differential equation solution gives y(x)= 1 c 1 + 3 x 2 3 , which I can use y ( 0 ) to find c 1 but then I'm not sure how to get y from that since the equation doesn't have any y's in it.
A similar question I found on this site showed to rearrange and integrate each side like d y y 4 = 2 x d x which gives 1 3 y 3 + c 1 = x 2 + c 2 , but then I don't know how to get y ( x ) from that frunction, it seems further from the solution than my first try.

Answer & Explanation

Kaylie Mcdonald

Kaylie Mcdonald

Expert

2022-07-08Added 19 answers

That's just a simple manipulation:
1 3 y 3 + c 1 = x 2 + c 2 1 3 y 3 = x 2 + c 2 c 1 1 3 y 3 = x 2 + c 1 c 2 y 3 = 1 3 x 2 + 3 ( c 1 c 2 ) y = 1 3 x 2 + 3 ( c 1 c 2 ) 3 .
Now it's just note that 3 ( c 1 c 2 ) is constant, say k, and we obtain
y = 1 3 x 2 + k 3 .

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