Given an IVP Fundamental Existence Theorem for Linear Differential Equations d"y da" du ly dan-1 d'y dz +an-1(2) y(xo) yo, y'(x) = ₁,, y(n-1)(o)=Y₁-1 If the coefficients a, (2),..., ao (x) and the right hand side of the equation g(x) are continuous on an interval I and if a, (2) #0 on I then the IVP has a unique solution for the point 20 € I that exists on the whole interval I. Consider the IVP on the whole real line +...+ a(z) d³y dr3 + ao(r)y= g(x) dr 1 dy (x² - 64) +2¹. + 2² +64 dz+y=sin(x) y(-1) = 582, y'(-1) = 16, y"(-1)=2, y"(-1) = 1, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
Given an IVP Fundamental Existence Theorem for Linear Differential Equations d"y da" du ly dan-1 d'y dz +an-1(2) y(xo) yo, y'(x) = ₁,, y(n-1)(o)=Y₁-1 If the coefficients a, (2),..., ao (x) and the right hand side of the equation g(x) are continuous on an interval I and if a, (2) #0 on I then the IVP has a unique solution for the point 20 € I that exists on the whole interval I. Consider the IVP on the whole real line +...+ a(z) d³y dr3 + ao(r)y= g(x) dr 1 dy (x² - 64) +2¹. + 2² +64 dz+y=sin(x) y(-1) = 582, y'(-1) = 16, y"(-1)=2, y"(-1) = 1, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Given an IVP
Fundamental Existence Theorem for Linear Differential Equations
dry
da"
+an-1(x).
d-ly
dr-1
y(To) = yo, y' (To) = ₁, ..., y(n-1) (to) = Y₁-1
If the coefficients a, (2),..., ao (x) and the right hand side of the equation g(x) are continuous on an interval I and if a, (2) #0 on I then the IVP has a unique solution for the
point o I that exists on the whole interval I
Consider the IVP on the whole real line
an(x).
dy
+...+ a₁(x). + ao(r)y = g(x)
dr
dy
1
(x² - 64) +zª. +
d'y d³y
dz4
dr3 x² +64 dx+y=sin(x)
y(-1) = 582, y' (-1) = 16, y"(-1) = 2, y"(-1) = 1,
The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
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