## Fundamentals of ODE by Xu, Labute.

By Xu, Labute.

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**Sample text**

Obviously, not any set of n vectors can span the vector space V . It will be seen that {v1 , v2 , · · · , vn } span the vector space V , if and only if they are linear independent. The set S = span(v1 , v2 , . . , vn ) = {c1 v1 + c2 v2 + · · · + cn vn | c1 , c2 , . . , cn ∈ R} consisting of all possible linear combinations of the vectors v1 , v2 , . . , vn form a subspace of V , which may be also called the span of {v1 , v2 , . . , vn }. Then V = span(v1 , v2 , . . , vn ) if and only if v1 , v2 , .

1). This DE is a logistic model for a population having 2 as the size of a stable population. The equation dy/dx = −y(2 − y)(3 − y) has three equilibrium states: y = 0, 2, 3. 2). The equation dy/dx = −y(2 − y)2 has two equilibrium states: y = 0, 2. 3). The sink is stable, source is unstable, whereas the node is semi-stable. The node point of the equation y = f (y) can either disappear, or split into one sink and one source, when the equation is perturbed with a small amount ε and becomes: y = f (y) + ε.

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 ). It can be shown that |yn − φ(xn )| ≤ Ch, where C is a constant which depends on p.