Second order ODEs- P.Integral for e^xsinx

In summary, the method of undetermined coefficients can be used to find the particular integral for a given combination of polynomials, exponentials, and sinusoidals. By setting the function f equal to a combination of exponential and trigonometric terms, the coefficients can be solved for using the given differential equation.
  • #1
Smith987
6
0
Hi guys, I really have no idea how to approach finding the particular integral for, say:

f'' + 5f' + f= e^x sinx

Could anyone help me? And for future reference how do you go about finding the PI for any combination of polynomials/exponentials/sinusoidals?

Thanks in advance for the help!
 
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  • #2
The PI would just the the product of the PI for ex and sinx
 
  • #3
Smith987 said:
Hi guys, I really have no idea how to approach finding the particular integral for, say:

f'' + 5f' + f= e^x sinx

Could anyone help me? And for future reference how do you go about finding the PI for any combination of polynomials/exponentials/sinusoidals?

Thanks in advance for the help!

Have you ever heard of the "Method of Undetermined Coefficients"? It is covered in introductory differential equations textbooks.

You might try setting

[tex] f = A e^x sin(x) + B e^x cos(x) [/tex]

and substituting it into your DE. Then collect like terms and see what values A and B have to be in order to make the resulting expression true. ( Hint: the combination of coefficients in front of the cosine terms will have to equal zero.)
 
  • #4
Aha thanks for the help guys :)
 

FAQ: Second order ODEs- P.Integral for e^xsinx

1. What is a second order ordinary differential equation (ODE)?

A second order ODE is a mathematical equation that involves a function and its first and second derivatives. It represents the relationship between a function and its rate of change.

2. What is the P.Integral for e^xsinx?

The P.Integral for e^xsinx is a method used to solve second order ODEs with a specific form. It involves transforming the equation into a simpler form and then using integration to solve for the function.

3. How is the P.Integral for e^xsinx used to solve ODEs?

The P.Integral for e^xsinx is used by first transforming the equation into a simpler form using substitution. Then, the equation is integrated to solve for the function. Finally, the inverse substitution is used to obtain the final solution.

4. What is the general solution for a second order ODE?

The general solution for a second order ODE is a family of functions that satisfies the given equation. It contains two arbitrary constants, which can be determined by applying initial or boundary conditions to obtain a particular solution.

5. Why is the P.Integral for e^xsinx useful in solving ODEs?

The P.Integral for e^xsinx is useful because it provides a systematic approach to solving second order ODEs with a specific form. It allows for a more efficient and organized solution compared to other methods such as variation of parameters or undetermined coefficients.

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