Properties of dierential linear systems (Sect. 7.4)
Review: n n linear dierential systems.
Fundamental solutions to homogeneous systems.
Existence and uniqueness of solutions to IVP.
The Wronskian of n solutions.
Review: n n linear dierential systems.
Den
Review Exam 3.
Sections 6.1-6.6.
5 or 6 problems.
50 minutes.
Laplace Transform table included.
Exam: November 11, 2008. Problem 2
Example
Use Laplace Transform to nd y solution of
y 2y + 2y = (t 2),
y (0) = 1,
y (0) = 3.
Solution: Compute the LT of the e
Equations with discontinuous sources (Sect. 6.4).
Dierential equations with discontinuous sources.
We solve the IVPs:
(a) Example 1:
y + 2y = u (t 4),
y (0) = 3.
(b) Example 2:
y +y +
5
y = b (t ),
4
y (0) = 0,
y (0) = 0,
b (t ) =
1, t [0, )
0, t [, ).
(c
Equations with regular-singular points (Sect. 5.5).
Equations with regular-singular points.
Examples: Equations with regular-singular points.
Method to nd solutions.
Example: Method to nd solutions.
Recall:
The point x0 R is a singular point of the equati
Power series solutions near regular points (Sect. 5.2).
We study: P (x ) y + Q (x ) y + R (x ) y = 0.
Review of power series.
Regular point equations.
Solutions using power series.
Examples of the power series method.
Review of power series.
Denition
The
Generalized sources (Sect. 6.5).
The Dirac delta generalized function.
Properties of Diracs delta.
Relation between deltas and steps.
Diracs delta in Physics.
The Laplace Transform of Diracs delta.
Dierential equations with Diracs delta sources.
Generaliz
The Euler equation (Sect. 5.4).
Overview: Equations with singular points.
We study the Euler Equation:
(x x0 )2 y + p0 (x x0 ) y + q0 y = 0.
Solutions to the Euler equation near x0 .
The roots of the indicial polynomial.
Dierent real roots.
Repeated roots
Systems of linear dierential equations (Sect. 7.1).
n n systems of linear dierential equations.
Second order equations and rst order systems.
Main concepts from Linear Algebra.
n n systems of linear dierential equations.
Remark: Many physical systems must
The Laplace Transform (Sect. 6.1).
The denition of the Laplace Transform.
Review: Improper integrals.
Examples of Laplace Transforms.
A table of Laplace Transforms.
Properties of the Laplace Transform.
Laplace Transform and dierential equations.
The Lapla
The Laplace Transform and the IVP (Sect. 6.2).
Solving dierential equations using L[ ].
Homogeneous IVP.
First, second, higher order equations.
Non-homogeneous IVP.
Solving dierential equations using L[ ].
Remark: The method works with:
Constant coecient
The Laplace Transform of step functions (Sect. 6.3).
Overview and notation.
The denition of a step function.
Piecewise discontinuous functions.
The Laplace Transform of discontinuous functions.
Properties of the Laplace Transform.
Overview and notation.
O
Review of Linear Algebra (Sect. 7.2)
The dot product of n-vectors.
The matrix-vector product.
A matrix is a function.
The inverse of a square matrix.
The determinant of a square matrix.
n n systems of linear algebraic equations.
The dot product of n-vecto
Review for Exam 2.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers:
Regular-singular points (5.5).
Euler dierential equation (5.4).
Power series solutions (5.2).
Variation of pa
Convolution solutions (Sect. 6.6).
Convolution of two functions.
Properties of convolutions.
Laplace Transform of a convolution.
Impulse response solution.
Solution decomposition theorem.
Convolution solutions (Sect. 6.6).
Convolution of two functions.
Pr