Prerequisites from Calculus

Albert Fässler ()
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Albert Fässler: Bern University of Applied Sciences

Chapter Chapter 1 in Fast Track to Differential Equations, 2021, pp 1-37 from Springer

Abstract: Abstract Let us try to find a function f with the following two properties: For all x ∈ R 1.1 $$ f^{\prime}(x) = f(x) $$ f ′ ( x ) = f ( x ) 1.2 $$ f(0) = C $$ f ( 0 ) = C (a) First we approach this problem geometrically by using a so-called direction field. The idea is to draw a large number of “compass needles” aligned according to the slopes given at the y-values where y = f(x). The horizontal lines are so-called isoclines, i.e. lines along which the slope f′(x) is constant. (b) Evidently, for each value of C, there exists a solution to the problem (1.1), (1.2). (c) For the special case C = 0, the solution is f(x) = 0 for all x ∈ R.

Date: 2021
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DOI: 10.1007/978-3-030-83450-0_1

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