Solve 2nd Order ODE y'' + 9y' = cot(3t)

[stanli121]

Homework Statement

y'' + 9y' = cot(3t)

Homework Equations

The Attempt at a Solution

This is a linear second order ODE where y(t) and that's what I'm solving for. Can this be solved via an integrating factor or does the cot(3t) part make that invalid? Any help is appreciated.

 

[Dustinsfl]

You can solve the complementary homogeneous equation and then the nonhomogeneous equation.

$y''+9y'=0\Rightarrow m^2+9m=0\Rightarrow m(m+9)=0$ $m=0,-9$

$$y_c=C_1+C_2e^{-9t}$$

Now to form the $y_p$ equation, you need to identify what annihilates cos(3t). Do you know what does?

 

[LCKurtz]

You can solve the complementary homogeneous equation and then the nonhomogeneous equation.

$y''+9y'=0\Rightarrow m^2+9m=0\Rightarrow m(m+9)=0$ $m=0,-9$

$$y_c=C_1+C_2e^{-9t}$$

Now to form the $y_p$ equation, you need to identify what annihilates cos(3t). Do you know what does?

Good luck on that, that's a cotangent, not a cosine. I would suggest variation of parameters.

 

[stanli121]

I don't really know how to solve this equation. My first (and only) attempt at the solution is an integrating factor and that doesn't work. I'm not well versed in differential equations because I haven't gotten there yet in the math sequence.

 

[LCKurtz]

I don't really know how to solve this equation. My first (and only) attempt at the solution is an integrating factor and that doesn't work. I'm not well versed in differential equations because I haven't gotten there yet in the math sequence.

You are probably getting a bit ahead of yourself. Integrating factors are used in the first order equations. Your equation is second order and is solved by methods in the "Constant Coefficient" section of your text. There you will learn about solutions to the homogeneous equation and how to find particular solutions to the non-homogeneous equation.