Lecture 6 Notes

Lecture 6 Notes - Today Im out of town Tuesday(Jan 28 no...

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Today • I’m out of town Tuesday (Jan 28) • no office hours, no lecture, • read Variations of Parameters (3.6) - for interest, not on the exam. • The geometry of homogeneous and nonhomogeneous matrix equations • Solving nonhomogeneous equations • Method of undetermined coefficients

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Second order, linear, constant coeff, non homogeneous (3.5) • Our next goal is to figure out how to find solutions to nonhomogeneous equations like this one: • But first, a bit more on the connections between matrix algebra and differential equations . . . y 6 y + 8 y = sin(2 t )
Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form A x = 0

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Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form • A non-homogeneous matrix equation has the form A x = b A x = 0
Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form • A non-homogeneous matrix equation has the form • A homogeneous differential equation has the form A x = b A x = 0 L [ y ] = 0

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Some connections to linear (matrix) algebra • A homogeneous matrix equation has the form • A non-homogeneous matrix equation has the form • A homogeneous differential equation has the form • A non-homogeneous differential equation has the form A x = b A x = 0 L [ y ] = 0 L [ y ] = g ( t )
Solutions to homogeneous matrix equations • The matrix equation could have (depending on A) (A) no solutions. (B) exactly one solution. (C) a one-parameter family of solutions. (D) an n-parameter family of solutions. A x = 0 Choose the answer that is incorrect .

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Solutions to homogeneous matrix equations • The matrix equation could have (depending on A) (A) no solutions. (B) exactly one solution. (C) a one-parameter family of solutions. (D) an n-parameter family of solutions. A x = 0 Choose the answer that is incorrect .
Solutions to homogeneous matrix equations • The matrix equation could have (depending on A) (A) no solutions. (B) exactly one solution. (C) a one-parameter family of solutions. (D) an n-parameter family of solutions. A x = 0 Choose the answer that is incorrect . Possibilities:

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Solutions to homogeneous matrix equations • The matrix equation could have (depending on A) (A) no solutions. (B) exactly one solution. (C) a one-parameter family of solutions. (D) an n-parameter family of solutions. A x = 0 Choose the answer that is incorrect . Possibilities: x = 0
Solutions to homogeneous matrix equations • The matrix equation could have (depending on A) (A) no solutions. (B) exactly one solution. (C) a one-parameter family of solutions. (D) an n-parameter family of solutions. A x = 0 Choose the answer that is incorrect . Possibilities: x = C 1 1 1

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Solutions to homogeneous matrix equations • The matrix equation could have (depending on A) (A) no solutions. (B) exactly one solution.
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