# How many solutions does the differential equation have?

#### shashank dwivedi

The Initial Value Problem yâ€²=y^2/3, y(0)=0 has ?

(a) No solution.

(b) Unique solution.

(c) Two solutions.

(d) Infinitely many solutions.

As I understood, one solution is y=0 itself and other solution comes on solving the differential equation as 3y^(1/3)= x+c, on putting the initial conditions, I get the solution as y=(x^3)/27.
Hence my conclusion is the answer is option c) that is only two solutions. Am I correct?

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#### SDK

You can use the two solutions you have found to construct infinitely many solutions so the answer is (d). Try it to construct these yourself.

#### skipjack

Forum Staff
There are infinitely many piecewise-defined solutions.

One such solution is y = xÂ³/27 for x < 0, y = 0 for x in [0, 1], and y = (x - 1)Â³/27 for x > 1.

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#### SDK

Better just solve the equation rather than applying theorems of existence of solution of a DE.
https://faculty.math.illinois.edu/~tyson/existence.pdf
The classical existence/uniqueness theorem does not apply to the OP example because the derivative in the example isn't a Lipschitz function. That is precisely why the solutions aren't unique. But the theorem you linked here doesn't say anything at all about uniqueness because it doesn't apply at all.

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