Math 286 Midterm 1 Short Study Guide.pdf - Math 286 Midterm 1 Short Study Guide Things to know Know the basic vocabulary we’ve been using Some example

Math 286 Midterm 1 Short Study Guide.pdf - Math 286 Midterm...

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Math 286 Midterm 1 Short Study Guide Things to know: Know the basic vocabulary we ve been using. Some example are: Ordinary differential equations versus partial differential equations Order of an equation Linear versus nonlinear equations What it means for a function to be a solution to a differential equation Bernoulli equations, autonomous equations Linear independence, the Wronskian, and how they are related Integral curves, general solutions, initial conditions/initial value problems Be able to solve various kinds of ODES. This is (unsurprisingly) one of the more important topics. Some examples are: First order linear equations (method of integrating factors) Separable equations Solving equations using substitutions (if a problem like this appears, you will be told what substitution to use, but you need to be able to carry out the substitution correctly) Second order homogeneous linear equations with constant coefficients
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Unformatted text preview: (based on the characteristic polynomial. You will need to know how to handle the case of distinct, real roots; as well as the case of complex roots) Be able to glean information about solutions to ODEs without solving them. (Also one of the more emphasized topics.) Some examples of this are: • Identifying an ODE or its solutions from its slope field • Critical point and stability analysis of autonomous equations / bifurcation. • Applying the existence and uniqueness theorems for solutions to first order ODES (remember that we have two theorems of this type: one for linear equations and one for more general equations). Finding the largest interval of validity for linear equations also falls under here • Knowing the behavior of solutions to second order linear homogeneous equations with constant coefficients based off of the roots of the characteristic polynomial...
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