Explanation Given information: The every constant multiple of a solution is also a solution for a first-order differential equation, d y d t = f ( t , y ) , where f(t,y) is continuous on the entire ty plane. Calculation: Consider thedifferential equation, d y d t = f ( t , y ) , where f(t,y) is continuous on the entire ty plane. Let, y 1 be the solution of d y d t = f ( t , y ) , Then the differential equation becomes, d y 1 d t = f ( t , y 1 ) , The k be the arbitrary constant, it is mentioned that ky 1 is also a solution of, d y d t = f ( t , y )

4th Edition

ISBN: 9780495561989

Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 1.8, Problem 34E

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To calculate: To describe the behavior of the differential equation, dydt=f(t,y)

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Chapter 1.8, Problem 33E

Chapter 1.9, Problem 1E

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(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

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Differential Equations

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To describe the behavior of the differential equation, d y d t = f ( t , y )

Solution Summary: The author explains the behavior of the differential equation, which does not depend on the independent variable.

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To calculate: To describe the behavior of the differential equation, dydt=f(t,y)

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