mL^(2)θ’’ + cθ’ + mgLθ + kLθ = Asin(bt)

Ynh= Asin(bt) + Bsin(bt)

Help need to find the derivatives of Ynh and sub into the DE then use the method of Undetermined coefficients.

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- Jun 18th 2018, 06:43 AMtoddjames99Need help solving this non homogenous differential equation
mL^(2)θ’’ + cθ’ + mgLθ + kLθ = Asin(bt)

Ynh= Asin(bt) + Bsin(bt)

Help need to find the derivatives of Ynh and sub into the DE then use the method of Undetermined coefficients. - Jun 18th 2018, 07:56 AMWalagasterRe: Need help solving this non homogenous differential equation
- Jun 18th 2018, 08:04 AMtoddjames99Re: Need help solving this non homogenous differential equation
y'nh= Acos(bt) *b - Bsin(bt) *b

y''nh= -Asin(bt) * b^(2) - Bcos(bt) * b^(2) - Jun 18th 2018, 08:29 AMtoddjames99Re: Need help solving this non homogenous differential equation
mL^2(-Asin(bt)*b^2-Bcos(bt)*b^2) + C(Acos(bt)*b-Bsin(bt)*b) + mgL(Asin(bt) + Bcos(bt)) + kL(Asin(bt) + Bcos(bt)) = Asin(bt)

- Jun 18th 2018, 09:22 AMromsekRe: Need help solving this non homogenous differential equation
do you mean

$Y_{nh} = A \cos(bt) + B \sin(bt)$ ? - Jun 18th 2018, 09:28 AMtoddjames99Re: Need help solving this non homogenous differential equation
Yes exactly

- Jun 18th 2018, 10:05 AMromsekRe: Need help solving this non homogenous differential equation
I used $c_1,~c_2$ instead of $A,~B$ for dummy coefficients in $Y_{nh}$ since there is a non-dummy coefficient labelled $A$ in the differential equation.

Attachment 38799 - Jun 18th 2018, 10:18 AMWalagasterRe: Need help solving this non homogenous differential equation
Your original equation can be written$$

mL^2\Theta'' +c\Theta' +(mgL+kL)\Theta = A\sin(bt)$$It will simplify things if you rename the complicated expressions, for example call $u = mL^2,~v =mgL + kL$ so your equation becomes$$

u\Theta'' + c \Theta' + v\Theta = A\sin(bt)$$You can put the $u$ and $v$ back in at the end. Now, when you try for a particular solution, you don't want to use an $A$ because it will be confused with the $A$ on the right side. So try something like $P\cos(bt) + Q\sin(bt)$ for a particular solution. That way you won't get the capital $C$ or $B$ letters confused with the $c$ and $b$ that are already in the equation. Try that.