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Unformatted text preview: Week 6 Differential Equations Summary Introduction and Terminology Differential equation: A differential equation (DE) is an equation involving a variable t , a function y ( t ), and its derivative y ( t ) (or some higher derivative like y 00 ( t )). For instance: y ( t ) = 2 y ( t ) + 3 , y = t 3 y 2 , y = e t 1 + y 2 . The study of DEs is a large and important part of mathematics, and in this course we will only scratch its surface. We will see examples like the ones above, but there are many other forms that we will not see, like cos( y ) = y 2 + t or y 00 = 2 y + 3. All DEs in this course will be first-order and separable , which we will define below. First-order: A DE is called first-order if the only derivative it involves is y (so no higher derivatives like y 00 ). Linear: A first-order DE is called linear if we can put it in the form y = ay + b . Solution: A solution to a DE is a function y ( t ) that satisfies the equation. For example, y ( t ) = e 2 t is a solution of y = 2 y , because y ( t ) = ( e 2 t ) = 2 e 2 t = 2 y ( t ). General solution: The general solution of a first-order DE is a function that depends on a constant C and gives all possible solutions. For example, y ( t ) = C e 2 t is the general solution of y = 2 y . Note : An antiderivative y ( t ) of a function f ( t ) is a solution to the DE y ( t ) = f ( t ). The general antiderivative (aka R f ( t ) dt ) is the general solution y ( t ) + C to that DE. Initial Value Problem: An initial value problem (IVP) is a DE together with an initial condition , which gives one value of the function. For instance: y ( t ) = 2 y ( t ) , y (0) = 3 ....
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