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Unformatted text preview: LECTURE NOTES Math 16C Short Calculus Spring 2008, Section 2 Dr. Peter Malkin May 27, 2008 1 Section C.1: Solutions of differential equations Points to cover (Section C.1): • What is a differential equation? • What is a general solution of a D.E.? • What is a particular solution of a D.E.? Definition 1 A differential equation is an equation involving a differentiable function (e.g. y = f ( x ) ) and one or more of its derivatives (e.g. y ′ = dy dx and y ′′ = d 2 y dx 2 ). E.g. y ′′ − 4 y ′ + 3 y = 0 or equivalently d 2 y dx 2 − 4 dy dx + 3 y = 0 ( ∗ ). Definition 2 A solution of a D.E. is a function y = f ( x ) that satisfies it. E.g. y = 4 e x + 5 e 3 x is a solution of ( ∗ ). Question: Verify that y = 4 e x + 5 e 3 x is a solution of ( ∗ ). Answer: y ′ = dy dx = d dx bracketleftbig 4 e x + 5 e 3 x bracketrightbig = 4 d dx [ e x ] + 5 d dx bracketleftbig e 3 x bracketrightbig = 4 e x + 15 e 3 x. y ′′ = d 2 y dx 2 = d dx bracketleftbigg dy dx bracketrightbigg = d dx bracketleftbig 4 e x + 15 e 3 x bracketrightbig = 4 e x + 45 e 3 x . Then, putting y , y ′ and y ′′ into ( ∗ ), we have y ′′ − 4 y ′ + 3 y = bracketleftbig 4 e x + 45 e 3 x bracketrightbig − 4 bracketleftbig 4 e x + 15 e 3 x bracketrightbig + 3 bracketleftbig 4 e x + 5 e 3 x bracketrightbig = 4 e x + 45 e 3 x − 16 e x − 60 e 3 x + 13 e x + 15 e 3 x = [4 − 16 + 12] e x + [45 − 60 + 15] e 3 x = 0 . Is this the only solution of ( ∗ )? NO! The function y = − 2 e x + 11 e 3 x is also a solution of ( ∗ ). Check this yourselves. In fact, there are infinitely many different solutions of ( ∗ )! Consider y = Ae x + Be 3 x where A and B are constants. Then, dy dx = Ae x + 3 Be 3 x and d 2 y dx 2 = Ae x + 9 Be 3 x . So, y ′′ − 4 y ′ + 3 y = bracketleftbig Ae x + 9 Be 3 x bracketrightbig − 4 bracketleftbig Ae x + 3 Be 3 x bracketrightbig + 3 bracketleftbig 4 e x + 5 e 3 x bracketrightbig = Ae x + 9 Be 3 x − 4 Ae x − 12 Be 3 x + 3 Ae x + 3 Be 3 x = [ A − 4 A + 3 A ] e x + [9 B − 12 B + 3 B ] e 3 x = 0 . There is one solution of ( ∗ ) for every choice of A and B . The function y = Ae x + Be 3 x is called a general solution of the D.E. A particular solution is a solution where A and B are fixed. • A general solution of a D.E. has unspecified constants. • A particular solution of a D.E. has no unspecified constants. E.g. y = 4 e x + 5 e 3 x and y = − 2 e x + 11 e 3 x are particular solutions of ( ∗ ). 2 The particular solutions of a D.E. are obtained from the general solution and some initial condi tions on the general solution and its derivatives. We use the initial conditions to find the values of the constants in the general solution....
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This note was uploaded on 06/15/2011 for the course CAL 3 taught by Professor Smith during the Spring '11 term at Arkansas State.
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