2nd order ODE with Sin function.

[rizvi71]

Hey guys!
I'm trying to solve a 2nd order differential equation. I am quite familiar with the method of solving these equations like treat them like characteristic equation ODE.
but there's a question which I really want to solve.
question is d^2 X/dt^2 +dX/dt +sinX=0.
How should I solve this??

 

[3.1415926535]

Hey guys!
I'm trying to solve a 2nd order differential equation. I am quite familiar with the method of solving these equations like treat them like characteristic equation ODE.
but there's a question which I really want to solve.
question is d^2 X/dt^2 +dX/dt +sinX=0.
How should I solve this??

That is not a linear ode and mathematica can't solve it... Is it solvable?

 

[rizvi71]

Hmmmm... my teacher gave me an assignment to solve this. So are you implying that its unsolvable??

 

[3.1415926535]

Hmmmm... my teacher gave me an assignment to solve this. So are you implying that its unsolvable??

Maybe he meant to write sint instead of sinx. If that's the case, then the solution is easy to find. Otherwise, unless you use some kind of "trick" I don't see how you can solve it

 

[rizvi71]

could you please give me some kind of a hint??
If it is "sint" how can i solve it??

 

[3.1415926535]

could you please give me some kind of a hint??
If it is "sint" how can i solve it??

Sure I can(if it is sint)

\begin{array}\\x''+x'+sint=0\\
x(t)=x_0(t)+x_p(t)\\
x_0''+x_0'=0\\
r^2+r=0\Leftrightarrow r(r+1)=0\Leftrightarrow r_1=0, r_2=-1\\
x_0(t)=c_1e^{0t}+c_2e^{-t}=c_1+c_2e^{-t}\end{array}

Let x_p=Asint+Bcost
Then
\begin{array}\\x''_p+x'_p+sint=0\Leftrightarrow (Asint+Bcost)''+(Asint+Bcost)'+sint=0\\-Asint-Bcost+Acost-Bsint+sint=0\Leftrightarrow sint(-A-B+1)+cost(-B+A)=0
\\\begin{Bmatrix}
-A& -B& =-1\\
-B& +A& =0
\end{Bmatrix}\Leftrightarrow \begin{Bmatrix}
B& +B& =1\\
& A& =B
\end{Bmatrix}\Rightarrow 2B=1\Leftrightarrow B=\frac{1}{2}=A\\
x_p=\frac{1}{2}(sint+cost)
\end{array}

Therefore, x(t)=c_1+c_2e^{-t}+\frac{1}{2}(sint+cost)

 

[rizvi71]

thanx a lot buddy !
Really appreciate it !

 

[3.1415926535]

thanx a lot buddy !
Really appreciate it !

No problem. Just ask your teacher and tell me whether it is sint or sinx.

 

[rizvi71]

Yeah sure will.
In the mean time can you please work something out if it is sinx. I'm really curious about this question.