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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 noncontinuous 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.
 Spring '13
 EricCytrynbaum
 Differential Equations, Equations

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