Non-homogeneous 1st order diff equation

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In summary, the conversation is about solving a differential equation with a particular integral involving e^30t and a trigonometric function. The person has been able to get the complimentary function but is struggling with finding the particular integral. They have attempted to integrate by parts but have not been successful. They are seeking help and are not confident in their calculus skills.
  • #1
becca1989
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Homework Statement


Hi, I have to solve the following differential equation and while I can get the complimentary function I can't get the particular integral.

y'+30y=20sin(alpha*t)+alpha*cos(alpha*t)


Homework Equations



How do I integrate the product of e^30t and alpha*cos(alpha*t) in order to find the particular integral?

The Attempt at a Solution



I've got y=Ce^(-30t) as the complimentary function but can get no further. Any help would be brilliant, thanks!
 
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  • #2
Integrate by parts

u=e-30t

dv=20sin(αt)+αcos(αt) dt
 
  • #3
Thanks, yeah I've been trying to do it by parts for the last little while but it get's very messy, very quickly whenever I have to integrate by parts twice. I'm rubbish at calculus, I end up with the same solution but it's obviously even near being correct.
 

1. What is a non-homogeneous 1st order differential equation?

A non-homogeneous 1st order differential equation is an equation that involves an unknown function and its first derivative, along with a non-zero function on the right side of the equation. This non-zero function is what makes the equation "non-homogeneous."

2. How is a non-homogeneous 1st order differential equation different from a homogeneous one?

In a homogeneous 1st order differential equation, the non-zero function on the right side of the equation is equal to zero. This means that the equation does not contain any external forces or influences, and the solution will be a linear combination of exponential functions. In a non-homogeneous equation, the non-zero function introduces external forces or influences that affect the behavior of the solution.

3. What are the general methods for solving non-homogeneous 1st order differential equations?

The general methods for solving non-homogeneous 1st order differential equations are the variation of parameters method, the undetermined coefficients method, and the Laplace transform method. These methods involve manipulating the equation and applying specific techniques to find the solution.

4. Can non-homogeneous 1st order differential equations be solved analytically?

Yes, non-homogeneous 1st order differential equations can be solved analytically using the general methods mentioned above. However, in some cases, an exact analytical solution may not be possible, and numerical methods may be used instead.

5. What are some real-world applications of non-homogeneous 1st order differential equations?

Non-homogeneous 1st order differential equations have various applications in physics, engineering, and other fields. They can be used to model the behavior of electrical circuits, chemical reactions, population growth, and many other phenomena. They are also essential in the study of control systems and signal processing.

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