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notes 3-1

# notes 3-1 - t → ∞ EXAMPLE Solve the initial value...

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SECTION 3.1 SECOND ORDER LINEAR ODE’S WITH CONSTANT COEFFICIENTS The general second order linear ODE can be written in the form P ( t ) y 00 + Q ( t ) y 0 + R ( t ) y = G ( t ) . When G ( t ) = 0 (identically), we say the equation is homogeneous . Turns out that if we can solve homogeneous equations, then we can solve nonhomogeneous ones! We specialize further to the case where P, Q, and R are constant functions. We are lucky because (1) we can always solve the DE, and (2) lots of DE’s in applications are like this! EXAMPLE. Solve y 00 + 8 y 0 + 15 y = 0. EXAMPLE, CONTINUED. Now add the initial conditions y (0) = 3, y 0 (0) = 1. What happens to the solution as

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Unformatted text preview: t → ∞ ? EXAMPLE. Solve the initial value problem y 00-2 y-3 y = 0, y (0) = α , y (0) = 3. Then ﬁnd α so that the solution approaches 0 as t → ∞ . What happens if we are just a tiny tiny bit oﬀ when we start this system, that is, we start not with y (0) = α but with y (0) = α + ± ? Assume we continue to hit the desired y (0) on the nose. What if we get a repeated root? How do we know we have all the solutions? What if the roots are complex numbers? HOMEWORK: SECTION 3.1...
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notes 3-1 - t → ∞ EXAMPLE Solve the initial value...

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