By Xu, Labute.

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**Additional resources for Fundamentals of ODE, Edition: lecture notes**

**Example text**

1 Lemma If f (x1 , x2 , . . xn ) and its partial derivatives are continuous on an ndimensional box R, then for any a, b ∈ R we have |f (a) − f (b)| ≤ ∂f ∂f (c) + · · · + (c) ∂x1 ∂xn |a − b| where c is a point on the line between a and b and |(x1 , . . , xn )| = max(|x1 |, . . , |xn |). The lemma is proved by applying the mean value theorem to the function G(t) = f (ta + (1 − t)b). This gives G(1) − G(0) = G (c) for some c between 0 and 1. The lemma follows from the fact that G (x) = ∂f ∂f (a1 − b1 ) + · · · + (an − bn ).

Where xn+1 − xn = h > 0 is called the step size. In general, the smaller the value of the better the approximations will be but the number of steps required will be larger. We begin by integrating y = f (x, y) between xn and xn+1 . If y(x) = φ(x), this gives φ(xn+1 ) = φ(xn ) + xn+1 xn f (t, φ(t))dt. As a first estimate of the integrand we use the value of f (t, φ(t)) at the lower limit xn , namely f (xn , φ(xn )). Now, assuming that we have already found an estimate yn for φ(xn ), we get the estimate yn+1 = yn + hf (xn , yn ) for φ(xn+1 ).

Integrating Factors. If the differential equation M + N y = 0 is not exact it can sometimes be made exact by multiplying it by a continuously differentiable function µ(x, y). Such a function is called an integrating factor. An integrating ∂µN factor µ satisfies the PDE ∂µM ∂y = ∂x which can be written in the form ∂M ∂N − ∂y ∂x µ=N ∂µ ∂µ −M . ∂x ∂y This equation can be simplified in special cases, two of which we treat next. µ is a function of x only. This happens if and only if ∂M ∂y − ∂N ∂x N = p(x) is a function of x only in which case µ = p(x)µ.