Solve ODE using method of integrating factors.

In summary, the conversation discusses solving a specific equation using the method of integrating factors. The person attempts to get the equation in the form dy/dx + r(x)y = f(x) but struggles with identifying r(x) and f(x). They also try another method of making the equation an exact differential, but are unsure of how to determine the factor. Later, it is suggested to simply subtract y from both sides to find the solution y=sin(x).
  • #1
alias25
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Homework Statement


solve the following equation using method of intergrating factors:


Homework Equations


dy/dx = y + cosx - sinx


The Attempt at a Solution



i think i have to get it in the form dy/dx + r(x)y = f(x)

but i can't see how,

if i multiply or divide by a factor i won't have dy/dx on its own. and how do i identify what's f(x) and r(x)?

i also tried

dy + (-y -cosx +sinx)dx = 0

and then just finding a factor to make it an exact differntial, but I am not sure how i determine the factor.
 
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  • #2
simply subtract the y from both sides...
dy/dx -y = cosx - sinx

your function in front of y is simply a constant function (-1)
 
  • #3
its kinda obvious that y=sin(x) is a solution...
 

1. What is the method of integrating factors?

The method of integrating factors is a technique used to solve ordinary differential equations (ODEs). It involves multiplying both sides of the ODE by a suitable integrating factor, which helps to simplify the equation and make it easier to solve.

2. When is the method of integrating factors used?

The method of integrating factors is typically used when solving first-order linear ODEs, which are ODEs that can be written in the form y' + P(x)y = Q(x). It can also be used for some higher-order ODEs by reducing them to a system of first-order ODEs.

3. How does the method of integrating factors work?

The method of integrating factors works by multiplying both sides of the ODE by a suitable integrating factor, which is a function of the independent variable x. This factor is chosen in such a way that the left side of the equation becomes the derivative of the product of the integrating factor and the dependent variable y. This simplifies the equation, allowing it to be easily integrated.

4. What are the advantages of using the method of integrating factors?

The method of integrating factors can be very useful in solving ODEs because it can simplify complex equations and make them easier to solve. It also allows for the use of standard integration techniques, such as separation of variables or integration by parts, to find an explicit solution.

5. Are there any limitations to the method of integrating factors?

While the method of integrating factors is a powerful tool for solving ODEs, it is not always applicable. It can only be used for linear ODEs, and even then, it may not be the most efficient method for solving certain types of equations. In addition, finding the appropriate integrating factor can sometimes be a challenging task.

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