By Louis Brand
A direction in research dealing primarily with features of a true variable, this article for upper-level undergraduate scholars introduces the fundamental suggestions of their easiest atmosphere and proceeds with a number of examples, theorems said in a pragmatic demeanour, and coherently expressed proofs. 1955 edition.
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Additional resources for Advanced Calculus: An Introduction to Classical Analysis (Dover Books on Mathematics)
A requirement that the system be in some known state at time t = 0). For example, if we require u(0) = 1, then C = 1 and we obtain a unique particular solution u(t) = (t + 1)e−t . 4 is a plot of the one-parameter family of solutions for several values of C. 1; there, the word parameter refers to a physical number in the equation itself that is fixed, yet arbitrary (such as resistance R in a circuit). 8 Suppose we have an autonomous DE, say, u′ = 3u2 . If we know that both u = u1 (t) and u = u2 (t) are solutions, is the sum v(t) = u1 (t) + u2 (t) also a solution?
The solution curve passes through the point (0, 1), corresponding to the initial condition u(0) = 1. Re-emphasizing, the initial condition selects one of the many solutions of the DE; it fixes the value of the arbitrary constant C. There are many interesting mathematical, or theoretical, questions about initial value problems. 1. (Existence) Given an initial value problem, must there always be a solution? This is the question of existence. Note that there may be a solution even if we cannot find a formula for it.
We first separate variables in the DE to get (2u + 1)u′ = 1, and then integrate both sides with respect to t to obtain (2u + 1)u′ dt = 1dt. But u = u(t) and du = u′ (t)dt, and therefore (2u + 1)du = 1dt. Carrying out the antidifferentiation, or integration, while introducing an arbitrary constant C, we get the general implicit solution u2 + u = t + C. The initial condition u(0) = 1 translates to u = 1 at t = 0. Substituting into the solution formula gives C = 2. So the implicit solution is u2 + u = t + 2.
Advanced Calculus: An Introduction to Classical Analysis (Dover Books on Mathematics) by Louis Brand