Find the general solution for a nonhomogenous equation.

In summary, the conversation is about finding the general solution for a nonhomogenous equation. The equation given is y''+4y=3csc2t, which is then simplified to y''+4y=(3)/(sin(2t)). The individual then proceeds to guess y=Asint+Bcost and solves for A and B using variations of parameters. They also mention using trig identities to simplify the equation. They ask for suggestions on resources for learning variations of parameters and are directed to Wikipedia and a math tutorial website. Eventually, they reach a solution that makes sense to them.
  • #1
cyturk
8
0

Homework Statement


Find the general solution for a nonhomogenous equation.
y''+4y=3csc2t

Homework Equations


The Attempt at a Solution


I simplified the equation to...
y''+4y=(3)/(sin(2t))

Then I guessed...
y=Asint+Bcost
y'=Acost-Bsint
y''=-Asint-BcostThen I got...
-Asint-Bcost+4Asint+4Bcost=(3)/(sin(2t))

Simplifies to...
3Asint+3Bcost=(3)/(2sintcost))

NOTE: Trig Identity: sin(2t)=2sintcost

I am stuck and I am not even sure if I took the correct steps.
 
Last edited:
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  • #2
Variations of parameters.
 
  • #3
hunt_mat said:
Variations of parameters.

Thanks for your help! I have been trying to figure out Variation of Parameters and I can't seem to find it on Khan Academy. Any chance there are any videos or resources you can suggest?
 
  • #4
http://en.wikipedia.org/wiki/Variation_of_parameters" is always a good place as sometimes they'll give an example...like in what I posted.
 
Last edited by a moderator:
  • #6
Thanks for everyone's help! I got a solution, although it kinda looks weird, but it makes sense to me now! :)
 

FAQ: Find the general solution for a nonhomogenous equation.

What is a nonhomogenous equation?

A nonhomogenous equation is a type of differential equation where the right-hand side of the equation is a function that is not equal to zero. This means there is an external or non-zero force acting on the system.

How is a nonhomogenous equation different from a homogenous equation?

In a homogenous equation, the right-hand side is equal to zero, meaning there is no external force acting on the system. This makes it easier to solve compared to a nonhomogenous equation.

What is the general solution for a nonhomogenous equation?

The general solution for a nonhomogenous equation is the sum of the complementary function and the particular integral. The complementary function is the solution to the homogenous equation, while the particular integral is a specific solution to the nonhomogenous equation.

How do you find the complementary function for a nonhomogenous equation?

To find the complementary function, you first solve the homogenous equation by setting the right-hand side to zero. This will give you a general solution in terms of constants. These constants can then be determined by applying any initial conditions given in the problem.

Can you give an example of solving a nonhomogenous equation?

Sure, for example, the nonhomogenous equation y'' + 2y' + y = 3x + 5 can be solved by first finding the complementary function y_c = c1e^(-x) + c2xe^(-x). Then, the particular integral y_p = Ax + B can be found by substituting it into the original equation and solving for A and B. The general solution would then be y = c1e^(-x) + c2xe^(-x) + Ax + B.

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