# Second order non-linear differential equation modelling

I am having a bit of a problem while modelling a hydro-mechanical system, after a long hassle with fluid mechanics, I got this not so cute non-linear differential equation, I believe I have to linearize the equation in order to model it and I have to do so near a starting point. Here is the equation:

$$m x''(t)=-Bx' (t)- Kx(t)+ A_c p_3 (t)$$ and

$$(dp_3)/dt= A_c*(β/V_3) x'(t) ̇$$

substituting x'(t) in the first equation:

$$x''(t)=K_1p_3(t) - K_2p_3'(t)- K_3x(t)$$

I really hope you can help I've been battling this equation for a week now. Thanks a lot for your time Hijawi

• Is that your equation ? Am I correct ? – Isham Jan 12 '18 at 0:16
• Yes it is. I got it by substituting continuity equation of flow in momentum equation, normally the systems are linear and are easy to solve. I need to use something like Taylor expansion to linearize this function. – AbdelRahman Hijawi Jan 12 '18 at 2:11
• This is a linear equation already (if $K_1$, $K_2$, $K_3$ are constants)... What exactly is it that you want to do with it? And is that all you have? There are two unknown(?) functions $x(t)$ and $p_3(t)$ but only one equation. – Hans Lundmark Jan 12 '18 at 5:36
• Okay I will edit the post to show you how I got to my equations. – AbdelRahman Hijawi Jan 12 '18 at 11:06
• You can integrate the second equation to get $p_3(t)=a x(t)+b$, where $a$ is known and $b$ comes from your boundary conditions. Put this in first equation to get an easy 2nd order DE in $x(t)$. Maybe I am missing something, but this does not look to be a problem. – user121049 Jan 12 '18 at 13:02