22d+ + y² = xy (Bernoulli DE)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 32T
Related questions
Question
Can you solve this using the flow given for bernoulli
![Bernoulli's Equation
Sometimes it is possible to solve a nonlinear equation by making a change of the dependent
variable that converts it into a linear equation. In this case, we can solve such equation by
means of integrating factor method. The most important such equation has the form
dy
dx
(3)
where n € R, called the Bernoulli's equation or Bernoulli DE. If n = 0 or n = 1, then
(3) is linear.
Method of Solution: For n #1
If we multiply both sides of (3) by y", it becomes
Let uy-n. Then
=
+ R(x)y = S(x)y"
du
(1-n)y".
da
Multiply both sides of (4) by (1-n).
=
- dy
dx
1-n
+y¹-"R(x) = S(x)
dy
dx
dy
(1-n)y " +(1-n)y¹ R(x) = (1-n)S(x)
dx
The Bernoulli differential equation is now transformed into the form
du +(1-n)R(x) u = (1-n)S(x)
dx
The above equation is now linear in u and can be solved by integrating factor method.
du
(6)
d.x
+ P(x) u = f(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb25a34a-815b-4749-b20d-10674f4c7f14%2Fd875182e-2993-4136-99f8-87ad1c09525b%2F5fiewaj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Bernoulli's Equation
Sometimes it is possible to solve a nonlinear equation by making a change of the dependent
variable that converts it into a linear equation. In this case, we can solve such equation by
means of integrating factor method. The most important such equation has the form
dy
dx
(3)
where n € R, called the Bernoulli's equation or Bernoulli DE. If n = 0 or n = 1, then
(3) is linear.
Method of Solution: For n #1
If we multiply both sides of (3) by y", it becomes
Let uy-n. Then
=
+ R(x)y = S(x)y"
du
(1-n)y".
da
Multiply both sides of (4) by (1-n).
=
- dy
dx
1-n
+y¹-"R(x) = S(x)
dy
dx
dy
(1-n)y " +(1-n)y¹ R(x) = (1-n)S(x)
dx
The Bernoulli differential equation is now transformed into the form
du +(1-n)R(x) u = (1-n)S(x)
dx
The above equation is now linear in u and can be solved by integrating factor method.
du
(6)
d.x
+ P(x) u = f(x)
![4. 2.
dx
+ y² = xy (Bernoulli DE)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb25a34a-815b-4749-b20d-10674f4c7f14%2Fd875182e-2993-4136-99f8-87ad1c09525b%2Femlhtd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. 2.
dx
+ y² = xy (Bernoulli DE)
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