# The solution of ODE is not the same as the implicit equation it was used for.

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.

• Is there any further context? What is $φ$ relative to $p$? What is the unknown that you want to solve for? Can you give some more steps for the nebulously general "With this variable i can calculate the variables" and "For that i took the equation above and solved for $p$ as good as i can." – LutzL Sep 2 '18 at 7:46
• @LutzL $\varphi$ is the time dimension. $\varphi$ is not dependant on $p$. I want to solve the first equation for $p$. $\frac{dQ_{w}}{d\varphi}$ , $\frac{dU}{d\varphi}$ and $\frac{dQ_{b}}{d\varphi}$ are $f(p)$ But i ONLY solved $\frac{dU}{d\varphi}$ for $p$ and got an ODE. $\frac{dQw}{d\varphi}$ and $\frac{dQb}{d\varphi}$ are functions depending on $p$ but i did not further solve $\frac{dQw}{d\varphi}$ and $\frac{dQb}{d\varphi}$ for $p$ and just used the numerical values to solve the ODE. – desperate_ode_boi Sep 2 '18 at 8:29