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Find the particular solution. y' = ex − 8;    y(0) = 7

Find the particular solution. y' = ex − 8;    y(0) = 7

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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Find the particular solution.
y' = ex − 8;    y(0) = 7
Expert Solution
Step 1

The given differential equation is:

Calculus homework question answer, step 1, image 1

y(0)=7

 

Step 2

Take the Laplace on both sides of the equation.

Calculus homework question answer, step 2, image 1

Use the fact that:

Calculus homework question answer, step 2, image 2

 

Step 3

Apply the partial fraction method to decompose the denominator of

Calculus homework question answer, step 3, image 1

Into different factors.

Calculus homework question answer, step 3, image 2

Substitute s=0 to get the value of A.

Calculus homework question answer, step 3, image 3

Substitute s=1 to get the value of B.

Calculus homework question answer, step 3, image 4

Substitute the value of A and B in the partial fraction

Calculus homework question answer, step 3, image 5

Therefore the partial fraction of

Calculus homework question answer, step 3, image 6

is

Calculus homework question answer, step 3, image 7

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