please do these problems Which substitution can be used to solve the given equat

Question

please do these problems

Which substitution can be used to solve the given equations. Write down and determine the type of the equation obtained after this substitution. NO solving.

my work (not sure so pleae work them out for me):

for a is it okay if divided the whole thing by x^2 and use v=y/x vx=y y' v+xv' and so 2v/1-v^2=v+xv', homgenous. <is this right or not sure if right please sh

for b is it okay to divide the whole thing by y^2 and say it's bernouli taken v as y^-1 and v'=

also, do we have the same cases here?

Explanation / Answer

Yes you solved right.

a. Next step will be v+ xv' = 2v/(1- v^2)

v' + v/x = 2v/(x(1- v^2) and this is first order nonhomogeneous diff.eq..

B. 1/y= v

-y'/(y^2) = v'

Substitute.

-v' + 2v/t. = t

And yes this is homogeneous equation.

2.a. do same as 1st part of above question

Divide the whole thing by x2 , then y'= v + xv' = (3v +v2)/(1+v)

v' = (3v+v2 - v - v2)/(1+v) = 2v/(1+v) , first order nonhomogeneous.

B. Now let v = 1/y2

v' = -y'/(y3)

-v' +3t2v = t , homogeneous

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