Solving DE Homework: Non-Unique PI?

  • Thread starter neelakash
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In summary, the conversation discusses finding the general solution for a given differential equation and questioning the uniqueness of the particular integral. The general solution is provided and the need for further clarification on the definition of particular integral is mentioned. A suggestion to differentiate and solve for the remaining term in the equation is also given.
  • #1
neelakash
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1

Homework Statement



Given DE is D^2(y)+5Dy+4y=cos 2x ...where D=>(dy/dx)

We are to find the general solution.(I have done it already)
Is the particular integral unique?If not,find another PI and show that the general solution is the same for all PI.

Homework Equations


The Attempt at a Solution



I did the first part and got the general solution as

y=C1 exp(-x)+C2 exp(-4x)+(1/10)cos 2x

I cannot understand why the PI would be non-unique?Can anyone please check if I am correct in having that answer?
 
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  • #2
?? You might want to check the definition of "particular integral"! They are never unique.

As far as checking is concerned, all you have to do is differentiate! Did you notice that if y= (1/10)cos(2x) then Dy= (1/5)sin(2x) so 5 Dy= sin(2x). How are you going to get rid of that sin(2x) on the left side of the equation?
 

FAQ: Solving DE Homework: Non-Unique PI?

What are the steps for solving a non-unique DE homework?

To solve a non-unique DE homework, you will need to follow these steps:

  • 1. Identify the type of DE: Is it separable, homogeneous, exact, or Bernoulli?
  • 2. Check for uniqueness: Is the solution given unique or non-unique?
  • 3. Find the general solution: Use the appropriate method to find the general solution.
  • 4. Check for initial or boundary conditions: Use the given conditions to find the particular solution.
  • 5. Verify the solution: Substitute the particular solution into the original DE to ensure it satisfies the equation.

How do I determine if a given DE has a unique or non-unique solution?

You can determine the uniqueness of a solution by checking the coefficients of the DE. If the coefficients are continuous and bounded, then the solution is unique. However, if the coefficients are not continuous or unbounded, then the solution is non-unique.

Can I use the same method to solve both unique and non-unique DE homework?

Yes, you can use the same method to solve both unique and non-unique DE homework. The only difference is that for non-unique solutions, you need to add a constant of integration to the general solution.

What should I do if I encounter a non-unique solution while solving a DE homework?

If you encounter a non-unique solution while solving a DE homework, you should add a constant of integration to the general solution. This constant can be any real number and it will give you the complete set of solutions to the DE.

Are there any tips for solving DE homework with non-unique solutions?

Yes, here are some tips for solving DE homework with non-unique solutions:

  • 1. Always double-check your work to make sure you have found the correct general solution.
  • 2. When checking for initial or boundary conditions, be aware that there may be multiple solutions that satisfy the given conditions.
  • 3. If the given DE is separable, homogeneous, or exact, you can follow the same steps as you would for a unique solution, but remember to add a constant of integration to the general solution.
  • 4. If the given DE is Bernoulli, you will need to use a different method to solve for the general solution.

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