Solving Differential Equations
Workbook 20: Laplace Transforms 1. Solving ODEs using Laplace transforms We begin with a straightforward initial value problem involving a first order constant coefficient differential equation. Let us find the solution of dy dt +2y = 12e3t y(0) = 3 …
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Representing Periodic Functions by Fourier
learn.lboro.ac.ukFunctions by Fourier Series 23.2 Introduction In this Section we show how a periodic function can be expressed as a series of sines and cosines. We begin by obtaining some standard integrals involving sinusoids. We then assume that if f(t) is a periodic function, of period 2π, then the Fourier series expansion takes the form: f(t) = a 0 2 + X ...
Stationary Points 18 - Loughborough University
learn.lboro.ac.ukThe stationary value is f(0,2) = 0+24−16+5 = 13 Example 9 Find a second stationary point of f(x,y) = 8x2 +6y2 −2y3 +5. Solution f x = 16x and f y ≡ 6y(2 − y). From this we note that f x = 0 when x = 0, and f x = 0 and when y = 0, so x = 0, y = 0 i.e. (0,0) is a second stationary point of the function. It is important when solving the ...
The Exponential Form of a Complex Number 10
learn.lboro.ac.ukThe Exponential Form of a Complex Number 10.3 Introduction In this Section we introduce a third way of expressing a complex number: the exponential form. We shall discover, through the use of the complex number notation, the intimate connection between the exponential function and the trigonometric functions. We shall also see, using the ...
5. The Bernoulli Equation - Learn
learn.lboro.ac.ukNow apply this to this example: A reservoir of water has the surface at 310m above the outlet nozzle of a pipe with diameter 15mm. What is the a) velocity, b) the discharge out of the nozzle and c) mass flow rate. (Neglect all friction in the nozzle and the pipe). 3 .
6. The Momentum Equation - Loughborough University
learn.lboro.ac.ukIn fluid mechanics the analysis of motion is performed in the same way as in solid mechanics - by use of Newton’s laws of motion. Account is also taken for the special properties of fluids when in motion. The momentum equation is a statement of …
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