Given an IVP d"y an (x) dn-1 + an-1(x)- drn-1 dy +...+ a1 (x) + ao (2)y = g(x) dx y(To) = Yo, y'(ro) = Y1, ·…, y(a-1)(xo) = Yn–1 If the coefficients an (x), ..., ao(x) and the right hand side of the equation g(x) are continuous on an interval I and if an (x) +0 on I then the IVP has a unique solution for the point ro e I that exists on the whole interval I Consider the IVP on the whole real line d'y - 36) d'y dy + y = sin(x) 1 (22 +x4. + dr3 2² + 36 dx y(3) = 916, y'(3) = 14, y"(3) = 8, y"(3) = 8, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
Given an IVP d"y an (x) dn-1 + an-1(x)- drn-1 dy +...+ a1 (x) + ao (2)y = g(x) dx y(To) = Yo, y'(ro) = Y1, ·…, y(a-1)(xo) = Yn–1 If the coefficients an (x), ..., ao(x) and the right hand side of the equation g(x) are continuous on an interval I and if an (x) +0 on I then the IVP has a unique solution for the point ro e I that exists on the whole interval I Consider the IVP on the whole real line d'y - 36) d'y dy + y = sin(x) 1 (22 +x4. + dr3 2² + 36 dx y(3) = 916, y'(3) = 14, y"(3) = 8, y"(3) = 8, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![Given an IVP
\[
a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x) \frac{dy}{dx} + a_0(x) y = g(x)
\]
\[
y(x_0) = y_0, \quad y'(x_0) = y_1, \quad \ldots, \quad y^{(n-1)}(x_0) = y_{n-1}
\]
If the coefficients \(a_n(x), \ldots, a_0(x)\) and the right-hand side of the equation \(g(x)\) are continuous on an interval \(I\) and if \(a_n(x) \neq 0\) on \(I\) then the IVP has a unique solution for the point \(x_0 \in I\) that exists on the whole interval \(I\).
Consider the IVP on the whole real line
\[
(x^2 - 36) \frac{d^4 y}{dx^4} + x^4 \frac{d^3 y}{dx^3} + \frac{1}{x^2 + 36} \frac{dy}{dx} + y = \sin(x)
\]
\[
y(3) = 916, \quad y'(3) = 14, \quad y''(3) = 8, \quad y'''(3) = 8,
\]
The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval \(\underline{\phantom{I}}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee8fb823-9e56-4d34-a863-64debae68570%2Ff150cea6-3ef6-4be8-bbc9-17a58f985750%2F7mf8v1_processed.png&w=3840&q=75)
Transcribed Image Text:Given an IVP
\[
a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x) \frac{dy}{dx} + a_0(x) y = g(x)
\]
\[
y(x_0) = y_0, \quad y'(x_0) = y_1, \quad \ldots, \quad y^{(n-1)}(x_0) = y_{n-1}
\]
If the coefficients \(a_n(x), \ldots, a_0(x)\) and the right-hand side of the equation \(g(x)\) are continuous on an interval \(I\) and if \(a_n(x) \neq 0\) on \(I\) then the IVP has a unique solution for the point \(x_0 \in I\) that exists on the whole interval \(I\).
Consider the IVP on the whole real line
\[
(x^2 - 36) \frac{d^4 y}{dx^4} + x^4 \frac{d^3 y}{dx^3} + \frac{1}{x^2 + 36} \frac{dy}{dx} + y = \sin(x)
\]
\[
y(3) = 916, \quad y'(3) = 14, \quad y''(3) = 8, \quad y'''(3) = 8,
\]
The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval \(\underline{\phantom{I}}\).
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