Solving a Separable ODE: y'+ytanx=cosx with Initial Condition y(0)=1

In summary, the conversation discusses a problem involving separable ode's and integrating factor. The individual seeking help is unsure if they are on the right track, but others confirm that they have calculated the integrating factor correctly. The conversation ends with the individual expressing their gratitude for the help.
  • #1
eeriana
15
0

Homework Statement


y'+ytanx = cos x y(0)=1


Homework Equations





The Attempt at a Solution



We are studying separable ode's and integrating factor right now, I am a little confused... If someone could steer me in the right direction, it would be greatly appreciated... This is what I have so far:

P= tanx
[tex]\int[/tex]P = -ln|cosx|
[tex]\mu[/tex]=e[tex]^{}-lncosx[/tex]
[tex]\mu[/tex]= 1/cosx

(1/cosx*y)' =[tex]\int[/tex]1/cosx cosx

and this is where I get stuck... am I even on the right track?

Thanks

Eeriana
 
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  • #2
You are on the right track and have calculated the IF correctly and written the complete differential correctly. Note that you can simplify the integrand...
 
  • #3
But isn't 1/cosx * cosx = 1 Or am I having an algebraic malfunction?
 
  • #4
eeriana said:
But isn't 1/cosx * cosx = 1 Or am I having an algebraic malfunction?
Nope you are indeed correct.
 
  • #5
I thought I was doing something wrong... hmmm..now I am going to see if I can finish it!

Thanks for the help
 

Related to Solving a Separable ODE: y'+ytanx=cosx with Initial Condition y(0)=1

1. What is a separable ODE?

A separable ODE (ordinary differential equation) is a type of differential equation where the dependent variable and its derivative can be separated into two different functions. This allows for a simpler solution process.

2. How do you solve a separable ODE?

To solve a separable ODE, you first need to separate the dependent variable and its derivative into two different functions. Then, you can integrate both sides of the equation and solve for the constant of integration. Finally, you can use algebraic manipulation to solve for the dependent variable.

3. What is the importance of separable ODEs?

Separable ODEs are important because they are relatively easy to solve and can be used to model a wide range of physical phenomena, such as population growth, radioactive decay, and chemical reactions.

4. Can a separable ODE have multiple solutions?

Yes, a separable ODE can have multiple solutions. This is because the constant of integration can take on different values, resulting in different solutions to the equation.

5. Are there any limitations to using separable ODEs?

Although separable ODEs are useful for many applications, there are some limitations. They can only be used for first-order differential equations, and they may not always accurately model complex systems with multiple variables or non-linear relationships.

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