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Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550

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Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550
Chapter 9.1, Problem 3E
Textbook Problem
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(a) For what values of r does the function y = erx satisfy the differential equation 2y″ + y′ − y = 0?

(b) If r1 and r2 are the values of r that you found in part (a), show that every member of the family of functions y = a e r 1 x + b e r 2 x is also a solution.

(a)

To determine

To find: The values of r when the function y=erx satisfy the differential equation 2y+yy=0 .

Explanation of Solution

Given:

The function is y=erx .

The differential equation is 2y+yy=0 .

Calculation:

Obtain the values of r when the function y=erx satisfy the differential equation 2y+yy=0 .

Differentiate the function y with respect to x,

y=ddx(y)=ddx(erx)=rex

Thus, y=rerx .

Differentiate the function y with respect to x,

y=ddx(y)=ddx(rerx)=r2erx

Thus, y=r2erx .

Since the function y=erx satisfy the differential equation 2y+yy=0

Substitute the values of y,y and y in the differential equation 2y+yy=0 to compute the values of r as follows,

2y+yy=0

(b)

To determine

To show: Every member of the family of functions y=aer1x+ber2x is a solution of the differential equation 2y+yy=0

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Chapter 9 Solutions

Calculus: Early Transcendentals
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Ch. 9.1 - Explain why the functions with the given graphs...Ch. 9.1 - The function with the given graph is a solution of...Ch. 9.1 - Match the differential equations with the solution...Ch. 9.1 - Suppose you have just poured a cup of freshly...Ch. 9.1 - Psychologists interested in learning theory study...Ch. 9.1 - Von Bertalanffys equation states that the rate of...Ch. 9.1 - Differential equations have been used extensively...Ch. 9.2 - A direction field for the differential equation y'...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Use the direction field labeled I (above) to...Ch. 9.2 - Use the direction field labeled III (above) to...Ch. 9.2 - Sketch a direction field for the differential...Ch. 9.2 - Sketch a direction field for the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - Sketch the direction field of the differential...Ch. 9.2 - (a) Use Eulers method with each of the following...Ch. 9.2 - A direction field for a differential equation is...Ch. 9.2 - Use Eulers method with step size 0.5 to compute...Ch. 9.2 - Use Eulers method with step size 0.2 to estimate...Ch. 9.2 - Use Eulers method with step size 0.1 to estimate...Ch. 9.2 - (a) Use Eulers method with step size 0.2 to...Ch. 9.2 - The figure shows a circuit containing an...Ch. 9.3 - Solve the differential equation. 1. dydx=3x2y2Ch. 9.3 - Solve the differential equation. 2. dydx=xyCh. 9.3 - Solve the differential equation. 3. xyy=x2+1Ch. 9.3 - Solve the differential equation. 4. y+xey=0Ch. 9.3 - Solve the differential equation. 5. 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Brine that...Ch. 9 - One model for the spread of an epidemic is that...Ch. 9 - The Brentano-Stevens Law in psychology models the...Ch. 9 - The transport of a substance across a capillary...Ch. 9 - Populations of birds and insects are modeled by...Ch. 9 - Suppose the model of Exercise 22 is replaced by...Ch. 9 - Barbara weighs 60 kg and is on a diet of 1600...Ch. 9 - Find all functions f such that f' is continuous...Ch. 9 - A student forgot the Product Rule for...Ch. 9 - Let f be a function with the property that f(0) =...Ch. 9 - Find all functions f that satisfy the equation...Ch. 9 - Find the curve y = f(x) such that f(x) 0, f(0) =...Ch. 9 - A subtangent is a portion of the x-axis that lies...Ch. 9 - A peach pie is taken out of the oven at 5:00 pm....Ch. 9 - Snow began to fall during the morning of February...Ch. 9 - A dog sees a rabbit running in a straight line...Ch. 9 - (a) Suppose that the dog in Problem 9 runs twice...Ch. 9 - A planning engineer for a new alum plant must...Ch. 9 - Find the curve that passes through the point (3,...Ch. 9 - Recall that the normal line to a curve at a point...Ch. 9 - Find all curves with the properly that if the...Ch. 9 - Find all curves with the property that if a line...

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