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# MTH256 NOTES WEEK 0-1 MTH 256

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This 7 page Class Notes was uploaded by Natalie Notetaker on Thursday August 4, 2016. The Class Notes belongs to MTH 256 at Oregon State University taught by tbd in Summer 2016. Since its upload, it has received 8 views. For similar materials see Applied Differential Equations in Math at Oregon State University.

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Date Created: 08/04/16

MTH256:WEEK 0-1 Wednesday Week 0, September 21 , 2016 Basic Definitions: Differential Equation: equation involving an unknown function and its derivatives Order: “diffyQ” depends on the highest order derivative seen in the function Ordinary Differential Equation (ODE): involves a function of only one variable Solution: a function y=f(x) that satisfies the equation on some open interval Solution Curve: graph of the solution Integral Curve: For an ODE, it’s the graph of a relation where any segment of that graph that represents the graph of the ODE function is a solution curve Direction Fields aka Slope Fields: if f is defined on some rectangle R in the xy-plane, then a graph consisting of the line segments of the slope f(x,y) at each (x,y) is called a direction field The General Solution: expressions with arbitrary constants that you can impose conditions upon Ex. Given a function, verify that another function is a solution Friday Week 0, September 23 , 2016rd Linear First Order: a first order ODE is linear if it can be put into the form y’+p(x)y=f(x) aka, standard form. Otherwise it is not linear Homogeneous: obvious/trivial solution to linear ODE in standard form aka, we know y=0 Ex. For each of the equations, determine if it is linear. If so, determine if it is homogeneous Monday Week 1, Sept. 26 , 2016h Review and more examples! Special Cases: Also did an application problem called “Mixing Problems”: 1. Set up differential equation 2. Model IVP 3. Integration factor 4. Impose constraints Began discussing separable equations: Separable: a first order differential equation is separable if it can be written as h(y)y’=g(x) Wednesday Week 1, September 28 , 2016th *Pop Quiz Today!* worth 10 points Ex. Solve Each Separable Equation Mixing Problem Example: th Friday Week 1, September 30 , 2016 Proposition: Suppose g(x) is continuos on (a,b) and h(y) is continuous on (c,d). let G(x), H(y) be antiderivatives of g(x) and h(y) on each interval. Let x be a0point on (a,b) and y be a poi0t on (c,d) such that h(y 0 does not =0 and defines c=H(y )-G(x 0 0 Conclusion: Then there is a function y(x) defined on (a,b) contained in (a,b) such that H’(y(x))y’(x)=G’(x) Which is an implicit solution because sometimes you can’t solve all the way through for x an y, but you can solve for a relationship that makes it true

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