Section5.6Larsonnotes - 1 Section 5.6 Differential...

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Section 5.6: Differential Equations: Growth and DecayPractice HW from Larson Textbook (not to hand in)p. 364 # 1-7, 19, 25-34 Differential EquationsDifferential Equations are equations that contain an unknown function and one or more of its derivatives. Many mathematical models used to describe real-world problems rely on the use of differential equations.The differential equations we will study in this section involve the first order derivative and are of the form),(yxFy=Our goal will be to find a function )(xfy=that satisfies this equation. The following two examples illustrate how this can be done for a basic differential equation and introduce some basic terminology used when describing differential equations.Example 1: Find the general solution of the differential equation 23xy=
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Example 2: Find the particular solution of the differential equation .1)0(,32==yxy
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Finding Solutions Analytically – Separation of VariablesSuppose we are given the initial value problem.00)(),,(yxyyxFdxdyy===and suppose we can write dxdyas product of a function of xand a function of y.)()(xgyhdxdy=We can then find a solution, y(x), by “separating” the variables.Steps – Separation of Variables 1. Get all terms involving yon one side of the equation and all terms involving xon the other side.2.Integrate both sides.3.Solve for the solution y(x) (if possible).Some Useful Facts1.kkelogln=2.ueu=ln3.||ymeans -=0if0ifyyyyy3