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summaryoutline1 - 1 Problem Characteristics Algebraic vs...

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1 Problem Characteristics Algebraic vs. differential equations differential equations partial differential equations ordinary differential equations ordinary differential equations linear vs nonlinear ODEs order of ODE general solutions of an ODE the initial value problem for an ODE integral curves, general solution, and initial value problem 2 First Order ODEs 2.1 Linear First Order ODEs constant coefficient homogeneous form y + ay = 0 y = Ce - at nonhomogeneous form y + ay = b y ( t ) = Ce - at + b a general solution general explicit form y + p ( t ) y = g ( t ) μ ( t ) = ˜ Ce z ( t ) , z ( t ) = p ( t ) dt integrating factor y ( t ) = C μ ( t ) + 1 μ ( t ) μ ( t ) g ( t ) dt general solution IVP solution for general first order ODE assuming μ ( t 0 ) = 1 y ( t ) = 1 μ ( t ) y 0 + t t 0 μ ( s ) g ( s ) ds 1
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behavioral analysis of solutions asymptotic behavior identification of initial condition regions where the behavior changes qualitatively identification of values of t for which the solution has difficulties, e.g., singularity,
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