# Need help solving this non homogenous differential equation

• Jun 18th 2018, 06:43 AM
toddjames99
Need 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 AM
Walagaster
Re: Need help solving this non homogenous differential equation
Quote:

Originally Posted by toddjames99
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.

Please show us what you get for the derivatives and what happens when you plug them in so we can see what your difficulty is.
• Jun 18th 2018, 08:04 AM
toddjames99
Re: 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 AM
toddjames99
Re: 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 AM
romsek
Re: Need help solving this non homogenous differential equation
do you mean

$Y_{nh} = A \cos(bt) + B \sin(bt)$ ?
• Jun 18th 2018, 09:28 AM
toddjames99
Re: Need help solving this non homogenous differential equation
Yes exactly
• Jun 18th 2018, 10:05 AM
romsek
Re: 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 AM
Walagaster
Re: 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.