MATH251-Worksheet 1

MATH251-Worksheet 1 - MATH 251 Work sheet / Things to know...

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MATH 251 Work sheet / Things to know Chapter 1 1. Classifications of differential equation ODEs vs PDEs What is the order of an equation? What makes an equation linear ? Ex.1.1.1 Which equation is linear? What is each equation’s order? (a) y ″ − 5 t 3 y ′ + ln( t ) y = t e 2 t (b) y ″′ + 2 t y ′ = y 4 (c) y ″ + 3( y ′) 4 + sin(2 t ) y = 0 (d) y ′ + tan( t 2 ) y = f ( t ) Ex. 1.1.2 Can you give an example of a fourth order linear equation? A sixth order nonlinear equation? © 2011 Zachary S Tseng 1
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2. Direction fields What is it? How to draw it? Ex. 1.2.1 y ′ = t 2 + y What is an integral curve ? Trace the integral curve that passes through the point (1, 1). © 2011 Zachary S Tseng 2
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MATH 251 Work sheet / Things to know Chapter 2 1. First order linear equations Standard form: Ex. 2.1.1 y ′ + 9 y = 6 Some, as the above example, could be solve by “re-arrangement”, but there is a general method, that of integrating factor that solves this type of equations Know the formulas of the integrating factor method. (When using them, first be sure that the equation is in its standard form!) Integrating factor: μ ( t ) = General solution: y ( t ) = © 2011 Zachary S Tseng 3
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Ex. 2.1.2 t 3 y ′ + 4 t 2 y = 5 e −2 t Standard form is: μ ( t ) = y ( t ) = Ex. 2.1.3 (see Ex.1.2.1 ) y ′ = t 2 + y Standard form is: μ ( t ) = y ( t ) = Ex. 2.1.4 Use the integrating factor method to solve the equation in example 2.1.1. © 2011 Zachary S Tseng 4
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2. Initial Value Problems (I.V.P.) What are the 2 components of an IVP? Its solution(s), if exists, contains no arbitrary constant/coefficient: particular solution . Know the difference between a general solution
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MATH251-Worksheet 1 - MATH 251 Work sheet / Things to know...

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