18.03 Lecture 26: 18.03 Sep 30, 2015 (Lecture 9)
Document Summary
Use operator notation and understand the properties of linear time-intervariant (lti) operators. Understand how to use time-intervariance to solve a differential equation with a time-shifted input signal. Understand how to apply the exponential response formula (erf) to solve inhomogeneous odes with exponential inputs. Understand why the erf fails and how to apply the generalized erf in these situations. Solve most higher order, inhomogeneous ltis using the exponential response formula. Describe and apply the superposition principle for higher order, constant coef cient differential equations. A function (e. g. f(t) = t ) takes an input number and returns another number. An operator takes an input function and returns another function. For example, the differential operator / t takes an input function y(t) and returns y/ t. So , for instance (because of the chain rule) The operator d is linear, which means that. Because of this, d represents linear combinations, meaning that.
