Trouble finding particular soln to this diff eqn: y''(x) + (nπa/l)2 y(x) = Asin(x/l)

[christyful]

Trouble finding particular soln to this diff eqn: y''(x) + (nπa/l)2 y(x) = Asin(x/l)

This is my first time posting so sorry if I have this in the wrong place.

I've been stumped by the particular solution for this, I hope someone can help!

y''(x) + (nπa/l)² y(x) = Asin(x/l)

 

[Deleted member 4993]

This is my first time posting so sorry if I have this in the wrong place.

I've been stumped by the particular solution for this, I hope someone can help!

y''(x) + (nπa/l)² y(x) = Asin(x/l)

Then you have already obtained the homogeneous solution?

Can you please tell us what did you get?

 

[christyful]

Y_g = ACos(nπa/l) + BSin(nπa/l)

Thanks in advance for any help!

 

[christyful]

y_h = Acos(nπa/l) + Bsin(nπa/l)

 

[HallsofIvy]

The general solution to the associated homogeneous equation is (almost) as you say, \(\displaystyle y_h(x)= Bcos\left(\frac{n\pi ax}{l}\right)+ Csin\left(\frac{n\pi ax}{l}\right)\). (The "almost" is because you forgot the "x"! And you do not want to use "A" as one of the undetermined constants because it is used in the equation itself.)

The given right hand side is A sin(x/l). As long as \(\displaystyle n\pi a\) is NOT 1, you can use P sin(x/l)+ Q cos(x/l) as possible solution. If \(\displaystyle n\pi a= 1\) the general solution to the associated homogeneous solution would be \(\displaystyle y_h(x)= B cos(x/l)+ C sin(x/l)\) so you would need to try Px cos(x/l)+ Qx sin(x/l) as particular solution.