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22d+ + y² = xy (Bernoulli DE)

22d+ + y² = xy (Bernoulli DE)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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)
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)
Transcribed Image Text:4. 2. dx + y² = xy (Bernoulli DE)
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