Lecture 3 Notes

# Then in some interval t0 h t0 t0 h contained in

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Unformatted text preview: uniqueness (Section 2.4) ∂f Theorem 2.4.2 Let the functions f and be continuous in some ∂y rectangle α < t < β , γ < y < δ containing the point (t0 , y0 ). Then, in some interval t0 − h < t0 < t0 + h contained in α < t < β, there is a unique solution y = φ(t) of the IVP y ￿ = f (t, y ), y (t0 ) = y0 . • A couple questions/examples to explore on your own: Existence and uniqueness (Section 2.4) ∂f Theorem 2.4.2 Let the functions f and be continuous in some ∂y rectangle α < t < β , γ < y < δ containing the point (t0 , y0 ). Then, in some interval t0 − h < t0 < t0 + h contained in α < t < β, there is a unique solution y = φ(t) of the IVP y ￿ = f (t, y ), y (t0 ) = y0 . • A couple questions/examples to explore on your own: • Why don’t we get a solution all the way to the ends of the t interval? • Example: dy = y2 , dt y (0) = 1 Existence and uniqueness (Section 2.4) ∂f Theorem 2.4.2 Let the functions f and be continuous in some ∂y rectangle α < t < β , γ < y < δ containing the point (t0 , y0 ). Then, in some interval t0 − h < t0 < t0 + h contained in α < t < β, there is a unique solution y = φ(t) of the IVP y ￿ = f (t, y ), y (t0 ) = y0 . • A couple questions/examples to explore on your own: • Why don’t we get a solution all the way to the ends of the t interval? • Example: dy = y2 , dt y (0) = 1 dy √ = y, dt y (0) = 0 • How does a non-continuous RHS lead to more than one solution? • Example: Second order linear equations (Chapter 3) • The general form for a second order linear equation: y + p(t)y + q (t)y = g (t) ￿￿ ￿ Second order linear equations (Chapter 3) • The general form for a second order linear equation: y + p(t)y + q (t)y = g (t) ￿￿ ￿ Second order linear equations (Chapter 3) • The general form for a second order linear equation: y + p(t)y + q (t)y = g (t) ￿￿ ￿ • Now, an IVP requires two ICs: y (0) = y0 , y (0) = v0 ￿ Second order linear equations (Chapter 3) • The general form for a second order linear equation: y + p(t)y + q (t)y = g (t) ￿￿ ￿ • Now, an IVP requires two ICs: y (0) = y0 , y (0) = v0 ￿ • As with ﬁrst order linear equations, we have homogeneous (g=0) and nonhomogeneous second order linear equations. • We’ll start by considering the homogeneous case with constant coefﬁcients: ay + by + cy = 0 ￿￿ ￿ Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions, y1(t) and y2(t). This means that Homog. eq. with constant coeff. (Section 3.1) ay ￿￿ + by ￿ + cy = 0 • Suppose you already found a couple solutions,...
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## This note was uploaded on 02/12/2014 for the course MATH 256 taught by Professor Ericcytrynbaum during the Spring '13 term at The University of British Columbia.

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