Find Existence of Solution to Cauchy Problem in Space $x^2+y^2 \leq 4$

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In summary, the Picard theorem guarantees the existence and uniqueness of the solution to the Cauchy problem $y'=\cos x-1-y^2, y(0)=1$ on the interval $(-b,b)$ for any $b>0$.
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evinda
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Hello! (Wave)

We have the space $x^2+y^2 \leq 4$ and we consider the Cauchy problem $y'=\cos x-1-y^2, y(0)=1$.

I want to find for which $b>0$ the Picard theorem ensures the existence of the solution on $(-b,b)$.

I have thought the following:

Since $g(x,y):=\cos x-1-y^2$ is continuous as for $x$ in a space $A$ it remains to be Lipschitz as for $y$ in any closed and bounded subset of $A$, let $B=(b_1,b_2) \times [-c_1,c_1]$.

$|g(x,y_1)-g(x,y_2)|=|y_2^2-y_1^2|=|y_1-y_2| | y_1+y_2| \leq (|y_1|+|y_2|) |y_1-y_2|$.

Thus we can pick $c_1$ as $b$. Am I right?
 
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Hello! Yes, your reasoning is correct. According to the Picard theorem, if $g(x,y)$ is continuous and Lipschitz with respect to $y$ on a closed and bounded subset of the space $A$, then there exists a unique solution to the Cauchy problem $y' = g(x,y), y(0)=y_0$ on the interval $(-b,b)$, where $b>0$ and $y_0$ is the initial condition. In this case, $g(x,y) = \cos x - 1 - y^2$ is indeed continuous and Lipschitz with respect to $y$ on the closed and bounded subset $B=(b_1,b_2) \times [-c_1,c_1]$, where $b_1=0, b_2=2$ and $c_1=1$. Therefore, the Picard theorem ensures the existence and uniqueness of the solution on the interval $(-b,b)$ for any $b>0$. I hope this helps!
 

FAQ: Find Existence of Solution to Cauchy Problem in Space $x^2+y^2 \leq 4$

1. What is the Cauchy problem in mathematics?

The Cauchy problem is a type of initial value problem in mathematics that involves finding the solution to a differential equation with given initial conditions. It is named after French mathematician Augustin-Louis Cauchy.

2. What is the significance of finding the existence of a solution to the Cauchy problem?

Finding the existence of a solution to the Cauchy problem is important because it ensures that the differential equation has a unique solution for a given set of initial conditions. This is crucial for many applications in physics, engineering, and other fields.

3. How is the Cauchy problem related to the space $x^2+y^2 \leq 4$?

The space $x^2+y^2 \leq 4$ is a representation of a circular region with a radius of 2 units centered at the origin. The Cauchy problem in this space involves finding the solution to a differential equation within this circular region.

4. What methods are commonly used to determine the existence of a solution to the Cauchy problem?

Some common methods used to determine the existence of a solution to the Cauchy problem include the Picard-Lindelöf theorem, Gronwall's inequality, and the method of characteristics. These methods rely on different mathematical techniques such as fixed-point theorems, comparison principles, and integral equations.

5. Can the existence of a solution to the Cauchy problem be guaranteed for all initial conditions?

No, the existence of a solution to the Cauchy problem cannot be guaranteed for all initial conditions. In some cases, the differential equation may have no solution or multiple solutions for a given set of initial conditions. This is known as ill-posedness and it can occur due to the nonlinearity or complexity of the differential equation.

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