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45-48. General first-order linear equations Consider the general first-order linear equation y' (t) + a(t)y(t) = f(1). This equation can be solved, in principle, by defining the integrating factor P(t) = exp (f a(t) di). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes p(t)(y'(t) + a(t)y(t)) = (p(t)y(t)) = P(t)f(1). Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. y' () + - у() — 0, у(1) 3D 6 y(1) = 0, y(1) = 6

45-48. General first-order linear equations Consider the general first-order linear equation y' (t) + a(t)y(t) = f(1). This equation can be solved, in principle, by defining the integrating factor P(t) = exp (f a(t) di). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes p(t)(y'(t) + a(t)y(t)) = (p(t)y(t)) = P(t)f(1). Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. y' () + - у() — 0, у(1) 3D 6 y(1) = 0, y(1) = 6

Calculus: Early Transcendentals
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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45-48. General first-order linear equations Consider the general first-order linear equation y' (t) + a(t)y(t) = f(1). This equation can be solved, in principle, by defining the integrating factor P(t) = exp (f a(t) di). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes p(t)(y'(t) + a(t)y(t)) = (p(t)y(t)) = P(t)f(1). Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. y' () + - у() — 0, у(1) 3D 6 y(1) = 0, y(1) = 6
Transcribed Image Text:45-48. General first-order linear equations Consider the general first-order linear equation y' (t) + a(t)y(t) = f(1). This equation can be solved, in principle, by defining the integrating factor P(t) = exp (f a(t) di). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes p(t)(y'(t) + a(t)y(t)) = (p(t)y(t)) = P(t)f(1). Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. y' () + - у() — 0, у(1) 3D 6 y(1) = 0, y(1) = 6
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