How to you find the general solution of √1−4x^2y'=x?

engausidarb 2022-09-16 Answered
How to you find the general solution of 1 - 4 x 2 y = x ?
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Answers (1)

Kimberly Evans
Answered 2022-09-17 Author has 13 answers
Changing the notation from Lagrange's notation to Leibniz's notation we have:

1 - 4 x 2 d y d x = x which is a First Order separable DE which we can rearrange as follows:

d y = x 1 - 4 x 2 d x d y = x 1 - 4 x 2 d x

To integrate the RHS we need to use a substitution

Let u = 1 - 4 x 2 d u d x = - 8 x - 1 8 d u = x d x

Se we can now substitute and integrate to get our DE solution:

y = - 1 8 u d u
y = - 1 8 u - 1 2 d u
y = - 1 8 u 1 2 1 2 + C
y = - 1 4 u 1 2 + C
y = - 1 4 1 - 4 x 2 + C

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