# Nonhomogeneous: Undetermined coefficients

• andrewdavid
In summary, the conversation is about solving the equation (d^2x/dt^2)+(w^2)x=Fsin(wt), x(0)=0,x'(0)=0 using the method of undetermined coefficients. The first step is to solve the homogenous equation by assuming a solution of the form x(t)=e^{rt}. After substituting into the equation, the roots are found to be r=+/-wi. Then, a particular solution of the type y_p=Axsin(wt)+Bxcos(wt) is tried. The conversation ends with one person figuring out the solution with the help of the other.
andrewdavid
(d^2x/dt^2)+(w^2)x=Fsin(wt), x(0)=0,x'(0)=0

Hope that's readable. First part is second derivative of x with respect to t. w is a constant and F is a constant. I need to find a solution to this using method of undetermined coeffecients and I'm confused with all the different variables. Anyone get me started at least?

Well, first off start by solving the homogenous equation to find the fundamental solution.

$$\ddot{x} + \omega^{2}x = 0$$

After that try a Particular solution of the type

$$y_{p} = A x \sin(\omega t) + B x\cos(\omega t)$$

Remember that if the fundamental solution has already sin and cos, you will need to try a xsin and xcos, like this case.

Last edited:
I got my homogenous equation x''+(w^2)x=0 but I can't find my roots with that w^2 in there.

What seems to be the problem? Show me your work.

Here, i will start you off

$$\ddot{x} + \omega^{2}x = 0$$

we assume a as a solution

$$x(t) = e^{rt}$$

So we substitute in our ODE

$$r^{2}e^{rt} + \omega^{2}e^{rt} = 0$$

so

$$e^{rt}(r^{2} + \omega^{2}) = 0$$

because $e^{rt}$ cannot be equal to 0

$$r^{2} + \omega^{2} = 0$$

which ends up as

$$r = \pm \omega i$$

Last edited:
I figured it out, thanks a lot for your help, I was just being dumb.

## 1. What is the concept of nonhomogeneous: undetermined coefficients?

Nonhomogeneous: Undetermined coefficients is a method used in differential equations to find a particular solution. It involves finding a function that satisfies the nonhomogeneous equation by assuming a form and solving for the coefficients.

## 2. How is the nonhomogeneous: undetermined coefficients method different from other methods?

The nonhomogeneous: Undetermined coefficients method is different from other methods, such as variation of parameters, because it relies on making an educated guess for the particular solution instead of using integrals or derivatives to find the solution.

## 3. When is the nonhomogeneous: undetermined coefficients method typically used?

The nonhomogeneous: Undetermined coefficients method is typically used when the nonhomogeneous equation has constant coefficients and the nonhomogeneous term is a polynomial, exponential, or trigonometric function.

## 4. What is the process for solving a nonhomogeneous equation using undetermined coefficients?

The process for solving a nonhomogeneous equation using undetermined coefficients involves identifying the form of the particular solution, plugging in the assumed form into the equation, and solving for the coefficients by equating the coefficients of the same terms on both sides of the equation.

## 5. Are there any limitations to using the nonhomogeneous: undetermined coefficients method?

Yes, the nonhomogeneous: Undetermined coefficients method can only be used for linear nonhomogeneous equations with constant coefficients. It also may not work for all types of nonhomogeneous terms, such as those that contain logarithmic or inverse trigonometric functions.

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