Obtaining Equations from Piecewise Function Graphs You may be asked to write a piecewise function, given a graph.
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We will see a couple of theorems that will tell us when we can solve a differential equation. We will also see some of the differences between linear and nonlinear differential equations.
Theorem 1 Consider the following IVP. So, just what does this theorem tell us? First, it tells us that for nice enough linear first order differential equations solutions are guaranteed to exist and more importantly the solution will be unique. We may not be able to find the solution but do know that it exists and that there will only be one of them.
This is the very important aspect of this theorem. Knowing that a differential equation has a unique solution is sometimes more important than actually having the solution itself!
Notice as well that the interval of validity will depend only partially on the initial condition. Example 1 Without solving, determine the interval of validity for the following initial value problem. So we will need to divide out by the coefficient of the derivative.
This will allow us to find all possible intervals of validity for the differential equation. This will guarantee that both functions are continuous everywhere in each interval.
So, the interval of validity for the initial value problem is. The first theorem required a linear differential equation. There is a similar theorem for non-linear first order differential equations. Here is the theorem.
Consider the following IVP. Unlike the first theorem, this one cannot really be used to find an interval of validity. So, we will know that a unique solution exists if the conditions of the theorem are met, but we will actually need the solution in order to determine its interval of validity.
Example 2 Determine all possible solutions to the following IVP. This differential equation is separable and is fairly simple to solve. There is also a third solution to the IVP.
In this last example we had a very simple IVP and it only violated one of the conditions of the theorem, yet it had three different solutions. This immediately illustrates a difference between linear and non-linear differential equations.Home; Calculators; Algebra II Calculators; Math Problem Solver (all calculators) System of Equations Calculator.
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Now that we know what piecewise functions are all about, it’s not that bad!
In this section we will give an in depth look at intervals of validity as well as an answer to the existence and uniqueness question for first order differential equations. - [Voiceover] By now we're used to seeing functions defined like h(y)=y^2 or f(x)= to the square root of x. But what we're now going to explore is functions that are defined piece by piece over different intervals and functions like this you'll sometimes view them as a piecewise, or these types of function definitions they might be called a piecewise function definition.