# Fundamental theorem on existence and uniqueness of solutions of differential equations

Given two differential equations

1) $$\frac{dy}{dx} = x^{\frac{2}{5}}$$

2) $$\frac{dy}{dx} = y^{\frac{2}{5}}$$

with the initial condition $$y(0) = 0$$ find which lacks uniqueness of the solution, and explain this via the fundamental theorem on existence and uniqueness of solutions of differential equations.

What i get:

On 1) Continuous around $$(0,0)$$ so there exist at least one solution, but i get the partial derivative of y to be 0 so not a unique solution, but the solution say it has a unique solution

On 2) Continuous around $$(0,0)$$ so there exist at least on solution, and partial derivate of y is equal to $$\frac{y^{\frac{7}{5}}}{7}$$ so a unique solution. The solution say this is not a unique solution.

Someone who can help me with this?

$$y'=f(y,x)$$
For the second case, you have continuity of the function and existence but the derivative is not continuous so you don't have uniqueness. You integrated the RHS instead of differentiating $$f(y,x)$$. $$\dfrac {\partial y^{2/5} }{\partial y}=\dfrac 25 y^{-3/5}$$