MTH 306 Final Exam Cheat Sheet PDF

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Cheat Sheet for Final Exam...

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Undetermined Coefficients y = yh + yp Exponential: yp = Ae2x yp’ = 2Ae2x yp’’ = 4Ae2x Trig: yp = Asinx+Bcosx yp’ = Acosx-Bsinx yp’’ = -Asinx-Bcosx Polynomial: yp = Ax2+Bx+C yp’ = 2Ax+B yp’’ = 2A To find yh: Set RHS =0 and solve using characteristic equation

m=mass, c=friction, k=spring constant

Underdamped: c2-4km0 (r1≠r2) Harmonic Motion: c=0, no oscillations F=kx A=

√c

2 1

2

+c 2

Geometric Multiplicity: # of linearly independent eigenvectors (# of free variables) Algebraic Multiplicity: # of repeated eigenvalues Defect: Algebraic mult. – Geometric mult. Complete Eigenvector: Alg. Mult. = Geom. Mult. Generalized Solution:

Stable

Unstable

Integrating Factor Given: y’+ p(x)y = f(x) 1) Find r(x): 2) Multiply everything by r(x) 3) LHS should now be product rule. Integrate both sides.

Bernoulli Substitution

ODE: derivatives are taken w.r.t. only one variable/ only one independent variable PDE: partial derivatives of several variables/ several independent variables Order: The order of the largest derivative that appears Linear: If dependent variables and their derivatives appear only as first powers, not multiplied together, no other functions of dependent variables appear Nonlinear: If conditions for linearity are not met it is nonlinear Homogeneous: All terms depend on dependent variable. Else, nonhomogeneous Constant Coefficients: if coefficients of linear equation are constant functions Autonomous: If equation does not depend on the independent variable

Singular & Ordinary Points

Root Test

Ratio Test |x-a|...