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ODE Cheat Sheet First Order Equations Separable Ry0(x) = f(x)g(y) dy g(y) = R f(x)dx+C Linear First Order y0(x)+p(x)y(x) = f(x) (x) = exp R x p(˘)d˘ Integrating factor. ( y)0= f Exact Derivative. Solution: y(x) = 1 (x) nR f(˘) (˘)d˘+ C Exact 0 = M(x;y)dx+ N(x;y)dy Solution: u(x;y) = const where du= @u @x dx+ @u @y dy @u @x = M(x;y); @u @y = N(x;y) Condition: My= Nx Non-Exact Form (x;y)(M



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18.085 :: Linear algebra cheat sheet :: Spring 2014 1. Rectangular m-by-nmatrices Rule for transposes: (AB)T = BT AT The matrix AT Ais always square, symmetric, and positive semi-definite.

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What is Linear Algebra Linear Algebra Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. Why do we study Linear Algebra? Provides a way to compactly represent & operate on sets of linear equations. In machine

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18.085 :: Linear algebra cheat sheet :: Spring 2014 1. Rectangular m-by-n matrices • Rule for transposes: (AB)T = B T AT • The matrix AT A is always square, symmetric, and positive semi-definite.

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