Lecture 9 Notes

Dx 11 x 41 y dt y introduction to systems of equations

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Unformatted text preview: x form: dx 2 = t x − y + cos(2t) dt dy 3 = x + 4 sin(t)y + t dt ￿ ￿ ￿2 ￿￿ ￿ ￿ ￿ dx t −1 x cos(2t) = + y 1 4 sin(t) y t3 dt Introduction to systems of equations • Any linear system can be written in matrix form: dx 2 = t x − y + cos(2t) dt dy 3 = x + 4 sin(t)y + t dt ￿ ￿ ￿2 ￿￿ ￿ ￿ ￿ dx t −1 x cos(2t) = + y 1 4 sin(t) y t3 dt • We’ll focus on the case in which the matrix has constant entries. And homogeneous, to start. For example, ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y or x = Ax ￿ Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y or x = Ax ￿ • Think of the unknown functions as coordinates (x(t), y (t)) of an object in the plane. Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y or x = Ax ￿ • Think of the unknown functions as coordinates (x(t), y (t)) of an object in the plane. • Ax gives the velocity vector of the object located at x. Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y or x = Ax ￿ • Think of the unknown functions as coordinates (x(t), y (t)) of an object in the plane. • Ax gives the velocity vector of the object located at x. Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y or x = Ax ￿ • Think of the unknown functions as coordinates (x(t), y (t)) of an object in the plane. • Ax gives the velocity vector of the object located at ￿￿ 2 x= 1 x. Introduction to systems of equations • Geometric interpretation - direction fields. ￿￿ ￿ ￿￿ ￿ dx 11 x = 41 y dt y or x =...
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