Differential equations - undetermined coefficients

In summary, to find a particular solution for the equation y'' + 2y' + 5y = e^x * sin(x), the characteristic equation is first solved to obtain the homogeneous equation. Then, the undetermined coefficients method is used to determine the particular solution, which involves making a guess and solving for the unknown coefficients. This method can be confusing, so referencing outside resources may be helpful.
  • #1
accountkiller
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Homework Statement


Find a particular solution of y'' + 2y' + 5y = e^x * sin(x)

Homework Equations


The Attempt at a Solution


Characteristic equation: r^2 + 2r + 5 = 0. Through quadratic formula, r1 = -1+2i and r2 = -1-2i, so my homogeneous equation is y_h = c1 * e^(-x) cos(2x) + c2 * e^(-x) * sin(2x).
Now, I need to use the undetermined coefficients method. How? I'm trying to figure out what exactly this undetermined coefficients method is, as my notes from lecture are confusing. And the e^x * sin(x) on the right looks like it would make things difficult.

I'd appreciate any guidance in the right direction, thanks!
 
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  • #2
hi mbradar2! :smile:

(try using the X2 and X2 icons just above the Reply box :wink:)

there's a good guide at http://en.wikipedia.org/wiki/Undetermined_coefficients" , beginning …
the method of undetermined coefficients, otherwise known as the Lucky Guess Method …

:biggrin:
 
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FAQ: Differential equations - undetermined coefficients

What is the general process for solving differential equations using undetermined coefficients?

The general process for solving differential equations using undetermined coefficients involves the following steps:

  • Find the general solution to the homogeneous equation
  • Assume a particular solution based on the form of the non-homogeneous term
  • Substitute the assumed solution into the original equation and solve for the undetermined coefficients
  • Add the particular solution to the general solution of the homogeneous equation to get the complete solution for the original equation

When can we use the method of undetermined coefficients to solve a differential equation?

The method of undetermined coefficients can be used to solve a differential equation when the non-homogeneous term can be expressed as a linear combination of known functions (e.g. polynomials, exponential functions, trigonometric functions).

What is the difference between undetermined coefficients and variation of parameters?

Undetermined coefficients and variation of parameters are both methods used to solve non-homogeneous linear differential equations. The main difference between the two is that undetermined coefficients assumes a particular solution based on the form of the non-homogeneous term, while variation of parameters uses a general solution to the homogeneous equation and a variation of constants to find the particular solution.

Can the method of undetermined coefficients be used for non-linear differential equations?

No, the method of undetermined coefficients can only be used for linear differential equations. For non-linear differential equations, other methods such as separation of variables or substitution must be used.

What are the limitations of using undetermined coefficients to solve differential equations?

The method of undetermined coefficients has some limitations, including:

  • It can only be used for linear differential equations
  • The non-homogeneous term must be expressible as a linear combination of known functions
  • If the non-homogeneous term is a solution to the homogeneous equation, the method will not work

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