Application of existence and uniqueness theorem

[find_the_fun]

Given the differential equation \(\displaystyle y'=4x^3y^3\) with initial condition \(\displaystyle y(1)=0\)determine what the existence and uniqueness theorem can conclude about the IVP.

I know the Existence and Uniquness theorem has two parts 1)check to see if the function is differentiable and 2)check to see if \(\displaystyle \frac{\partial f}{\partial y}\) is continious. If they both are true you can condlude there is only one solution (a unique solution). If not, then you can't conclude anything.

My problem is I don't understand which function I should be checking. Should I be checking the differential equation it self, or do I first need to solve and and then check the (a) solution?

 

[evinda]

Given the differential equation \(\displaystyle y'=4x^3y^3\) with initial condition \(\displaystyle y(1)=0\)determine what the existence and uniqueness theorem can conclude about the IVP.

I know the Existence and Uniquness theorem has two parts 1)check to see if the function is differentiable and 2)check to see if \(\displaystyle \frac{\partial f}{\partial y}\) is continious. If they both are true you can condlude there is only one solution (a unique solution). If not, then you can't conclude anything.

My problem is I don't understand which function I should be checking. Should I be checking the differential equation it self, or do I first need to solve and and then check the (a) solution?

The initial value problem is of the form:

$$\left\{\begin{matrix}
y'=f(x,y)\\
\\
y(x_0)=y_0
\end{matrix}\right.$$

Therefore, the function $f$ is equal to $4x^3y^3$.