Lets say i have a variable named $p$. With this variable i can calculate the variables $\frac{dU}{d\varphi}$ and $\frac{dQ_{w}}{d\varphi}$ and with equation shown below also $\frac{dQ_{b}}{d\varphi}$.
$$ \frac{dQ_{b}}{d\varphi}=\frac{dU}{d\varphi}+p\cdot\frac{dV}{d\varphi}+\frac{dQ_{w}}{d\varphi} $$
Now i manually manipulated the course of $\frac{dQ_{b}}{d\varphi}$ and want to see its effect on $p$. For that i took the equation above and solved for $p$ as good as i can. The result was an ODE
$$ \frac{dp}{d\varphi}=-\kappa_{1}\cdot p+\frac{\kappa_{2}-1}{V}*(\frac{dQ_{b}}{d\varphi}-\frac{dQ_{w}}{d\varphi}) $$
To test the equation i firstly did not manipulate any variables and just used the numerical values i got from the previous calculations
That means that i should get the variable $p$ that i used to calculate the variables $\frac{dQ_{b}}{d\varphi}$ and $\frac{dQ_{w}}{d\varphi}$ when i solve the ODE right ?
Well NO, i dont and i dont understand why.
Please mind that $\kappa_{1}$ and $\kappa_{2}$ are still dependant on $p$. I also did not resolve $\frac{dQ_{w}}{d\varphi}$ or $\frac{dQ_{b}}{d\varphi}$ further for $p$ (just used the numerical courses). The Solver i used was MATLAB's ode45(changing the solver didnt change anything)
So my idea is that it is because i didnt dissolve my variables further for $p$ and i neglected some derivatives like $\frac{\partial \frac{dQ_{w}}{d\varphi} }{\partial p}$ ?
Every help would be appreciated. I am so desperate i would even pay money if i was allowed to.