Thread: Need help solving this non homogenous differential equation

1. 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.

2. Re: Need help solving this non homogenous differential equation

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.

3. 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)

4. 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)

5. Re: Need help solving this non homogenous differential equation

do you mean

$Y_{nh} = A \cos(bt) + B \sin(bt)$ ?

Yes exactly

7. 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.

8. 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.