Lecture 6 - Second Order Equations So Far… We have been...

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Second Order Equations
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So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second order linear equations. (Sort of)
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Bad (or Good?) News Higher order equations are MUCH more difficult to solve than first order equations. So we start simple, and work our way up. Linear equations are hard. Nonlinear equations are crazy hard.
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Linear Second Order Equations The form of a second order linear equation: a 0 ( t ) a 1 ( t ) a 2 ( t ) g ( t ) y 00 y 0 y + + = Not simple enough! a 0 ( t ) a 1 ( t ) a 2 ( t ) y 00 y 0 y + + = 0 Still not simple enough! Homogenous equation.
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Linear Second Order Equations The form of a second order linear equation: a 0 ( t ) a 1 ( t ) a 2 ( t ) g ( t ) y 00 y 0 y + + = Not simple enough! a 0 ( t ) a 1 ( t ) a 2 ( t ) y 00 y 0 y + + = 0 Still not simple enough! y 00 y 0 y + + = 0 A B C
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Linear Second Order Equations The form of a second order linear equation: a 0 ( t ) a 1 ( t ) a 2 ( t ) g ( t ) y 00 y 0 y + + = Not simple enough! a 0 ( t ) a 1 ( t ) a 2 ( t ) y 00 y 0 y + + = 0 Still not simple enough! y 00 y 0 y + + = 0 Okay, that’s a good place to start. A B C
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Second Order Linear Homogenous Equations with Constant Coefficients y 00 y 0 y + + = 0 A B C Side note: We already know one specific solution to this equation. Equilibrium Solution! y = 0 Always check for equilibrium solutions!!!!
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Second Order Linear Homogenous Equations with Constant Coefficients y 00 y 0 y + + = 0 A B C We will consider solutions of the form: Why?
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