Fourth solution of y'=10/3 xy^{2/5}, y(0)=0?

hommequidort0h

hommequidort0h

Answered question

2022-09-21

Fourth solution of y = 10 3 x y 2 / 5 , y ( 0 ) = 0?
By inspection, we know that y = 0 is a solution. If we separate variables, we get another solution, y = x 10 / 3 . The book tells me this much, and asks me to find the other 2 solutions that differ on every open interval containing x = 0 and are defined on ( , ).
I found the third solution (by guessing) to be y = x 10 / 3 . The only other possibility i could think about is x = 0, but I'm pretty sure that's not valid since y′ would be undefined. What is the last solution? How do we know there are only 4 possible solutions (and not more)?

Answer & Explanation

Abagail Meyers

Abagail Meyers

Beginner2022-09-22Added 10 answers

Step 1
y = x 10 / 3 is not a solution. It is decreasing where it should be increasing and increasing where it should be decreasing.
You could have any one of y = 0 and y = x 10 / 3 for x 0, and any one of those for x < 0. That makes 4 different solutions.
Step 2
There are also solutions of the form y = { ( x 2 a 2 ) 5 / 3  for  x < a 0  for  a x b ( x 2 b 2 ) 5 / 3  for  x > b where a < 0 < b + , but these don't differ on every open interval containing 0.

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