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Clutch Cargo
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Homework Statement
I'm having problems with this IVP
dy/dx=y^4-x^4 and y(0)=7 I know the answer is yes but I just don't see how to get it.
It's only in your fifth post that you tell us that this problem is about "the existence and uniqueness theorem" and you think your textbook is being confusing?Clutch Cargo said:The existence and uniqueness theorem is exactly what this problem is about. As I have stated the book only gives one example and it is nothing like this problem.
I have however looked at the solutions manual for other similar problems and I find in each case the answer book is only concerned with whether f(x,y) and f'(x,y) are continuous and it totally ignored the initial value given (in this case the y(0)=7)
I am wondering if the textbook is intentionally trying to make this confusing or what.
The theorem applies to this particular problem. So...Clutch Cargo said:It says that given the initial value problem:
dy/dx=f(x,y) y(xo)=yo
assume that f and df/dy are continuous fuctions in a rectangle
R={(x,y):a<x<b, c<y<d}
that contains the point (xo,yo). Then the initial value problem has a unique solution Psi(x)
in some interval xo-epsilon<x<xo+epsilon where epsilon is a positive number.
Correct. This takes care of what the theorem refers to as f(x,y).Clutch Cargo said:I know that y'=y^4-x^4 is continuous
This doesn't make sense, please phrase it more carefully. To ensure that the IVP has a unique solution there are two functions which need to be verified as continuous. You've stated above that f(x,y) is continuous...what else do you need to check?Clutch Cargo said:dy'/dxy=0 which is contiuous and integral(dy,dx)=xy^4-X^5/5 which is continuous so it seems that the theorem holds true.
An initial value problem is a type of mathematical problem that involves finding the solution to a differential equation by specifying the initial conditions of the problem. This means that we are given the value of the dependent variable at a specific initial point, and we need to find the value of the dependent variable at other points.
To solve an initial value problem, you first need to identify the differential equation that describes the problem. Then, you need to integrate the equation to find the general solution. Finally, you can use the given initial conditions to determine the specific solution to the problem.
Some common techniques used to solve initial value problems include separation of variables, integrating factors, and substitution of variables. These techniques can be applied to different types of differential equations, such as first-order, second-order, and systems of differential equations.
Initial value problems are important in science because they allow us to model and understand real-world phenomena. Many physical, biological, and social systems can be described using differential equations, and solving initial value problems helps us predict and analyze the behavior of these systems.
Some real-world examples of initial value problems include predicting the growth of a population, modeling the spread of a disease, and analyzing the motion of a pendulum. These problems can be solved using differential equations and provide valuable insights into the behavior of complex systems.