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By Brannan J., Boyce W.
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Extra resources for Differential equations. An introduction to modern methods and applications
That approximate the values of the solution φ(t) at the points t1 , t2 , t3 , . . If, instead of a sequence of points, you need a function to approximate the solution φ(t), then you can use the piecewise linear function constructed from the collection of tangent line segments. That is, let y be given by Eq. (4) in [t0 , t1 ], by Eq. (6) in [t1 , t2 ], and in general by y = yn + f (tn , yn )(t − tn ) (11) in [tn , tn+1 ]. 2 to approximate the solution of the initial value problem 3 dy 1 + y = − t, dt 2 2 y(0) = 1 (12) on the interval 0 ≤ t ≤ 1.
The function μ(t) is called an integrating factor, and the main difficulty is to determine how to find it. We will introduce this method in a simple example, and then show that the method extends to the general equation (5). EXAMPLE 1 Solve the differential equation dy − 2y = 4 − t. dt (9) Plot the graphs of several solutions and draw a direction field. Find the particular solution whose graph contains the point (0, −2). Discuss the behavior of solutions as t → ∞. The first step is to multiply Eq.
This may be a widely accepted physical law, or it may be a more speculative assumption based on your own experience or observations. In any case, this step is likely not to be purely mathematical, but will require you to be familiar with the field in which the problem originates. 4. Express the principle or law in step 3 in terms of the variables you chose in step 1. This may be easier said than done. It may require the introduction of physical constants or parameters and the determination of appropriate values for them.