First order separable differential equations (d^2y)/(dx^2)+2(dy)/(xdx) = 0 Using change of variable equals to Z=(dy)/(dx) what is the 1st order separable differential equation for Z as function of x? and solve for y(x).

traffig75 2022-09-14 Answered
First order separable differential equations
d 2 y d x 2 + 2 d y x d x = 0
Using change of variable equals to Z = d y d x what is the 1st order separable differential equation for Z as function of x?
and solve for y(x).
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Answers (1)

Jonah Cooke
Answered 2022-09-15 Author has 7 answers
Let Z = d y d x . Then d 2 y d x 2 + 2 d y x d x = 0 can be written as
d Z d x + 2 Z x = 0.
Then d Z Z = 2 d x x
d Z Z = 2 d x x
So it's easy to get Z = C x 2 , where C is a constant. Then for Z = d y d x = C x 2 . We get the solution is y = C x + C 0 , where C 0 is a constant.

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