Solving y'=y-sin(2x): Integral of -sin(2x)e^x Stuck, Help!

  • Thread starter koolrizi
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In summary, the integrating factor for the given differential equation is e^x and the integral of -sin(2x)e^x can be solved by using integration by parts. However, it is possible to solve for the expression directly, resulting in the answer 1/5(sin(2x)+2cos(2x)). Just be careful with the details, such as using the correct integrating factor and including the boundaries of the LHS of the equation.
  • #1
koolrizi
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I found the integrating factor to be e^x for DE y'=y-sin(2x)
Now i am stuck at integral of -sin(2x)e^x
Can you help me with it? I tried using integration by parts and i get to following
integral(sin(2x)(e^x)=sin(2x)e^x-2cos(2x)e^x-4*integral(sin(2x)(e^x)

I am stuck. Help! Thanks
 
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  • #2
Notice that you have the expression "integral(sin(2x)(e^x))" on both sides of your equation. You can therefore solve for it directly, and be done.
 
  • #3
directly? I think i am not seeing something
 
  • #4
The answer to this DE is 1/5(sin(2x)+2cos(2x) and i am not seeing how to get to this from what i have
 
  • #5
OH ok i got it...wasnt seeing that lol...THanks
 
  • #6
also careful with your results. looks like you missed a couple small details. your integrating factor is actually $e^{-x}$ and you should have the boundaries of the LHS of your equation(unless they're supposed to be =0)
 

1. What is the first step in solving y'=y-sin(2x)?

The first step is to separate the variables by moving all terms involving y to one side of the equation and all terms not involving y to the other side. This will give you the equation y' - y = -sin(2x).

2. How do I solve for y?

To solve for y, you will need to use an integrating factor. This is the function e^x, which is multiplied by both sides of the equation. This will simplify the left side of the equation to (e^x)y' - (e^x)y = -(sin(2x)e^x).

3. What is the integrating factor?

The integrating factor is the function e^x, which is multiplied by both sides of the equation to simplify the left side of the equation.

4. How do I solve the integral of -sin(2x)e^x?

To solve this integral, you will need to use integration by parts. The formula for integration by parts is ∫u dv = uv - ∫v du. In this case, let u = e^x and dv = -sin(2x). You can then solve for v and du, plug them into the formula, and integrate to find the solution.

5. What is the final solution for y?

After solving the integral of -sin(2x)e^x, you will get an expression for y in terms of x. The final solution will be y = (c + 1/2)e^x - 1/4sin(2x)e^x, where c is the constant of integration.

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