The discussion centers on applying the Existence and Uniqueness Theorem to the initial value problem defined by the differential equation dy/dx = y^4 - x^4 with the initial condition y(0) = 7. Participants clarify that the function f(x,y) = y^4 - x^4 must be analyzed to determine the existence of a unique solution at the point (0,7). The derivative f'(x,y) = 4y^3 is computed, which is essential for applying the theorem. The conclusion emphasizes the need to reference the theorem's statement to ascertain the uniqueness of the solution.
PREREQUISITESStudents studying differential equations, educators teaching calculus concepts, and mathematicians interested in the application of the Existence and Uniqueness Theorem.
bravoman said:Homework Statement
Given the equation dy/dx = y^4 - x^4, y(0) = 7, determine whether the existence/uniqueness theorem implies that the given initial value problem has a unique solution.
Homework Equations
Existence/Uniqueness Theorem
The Attempt at a Solution
To my understanding, you must assign f(x,y) = y^4 - x^4 then derive f(x,y) in terms of y.
f'(x,y) = 4y^3
This is as far as I have gotten.
Ray Vickson said:What is the statement of the existence/uniqueness theorem? How does it apply in your case?
That isn't what Ray asked. What does the Existence and Uniqueness Theorem say?bravoman said:I need to use the theorem to determine of the equation dy/dx = y^4 - x^4 has a unique solution at point (0,7). Is that correct?