Electronic – Circuit Simulation Software

As far as I remember the SPICE algorithms you do not want its internal matrices dumped on you. Also I believe for transient analysis it converts all capacitors to voltage sources (and inductors to current sources) for every time step and solves the non-linear circuit like a DC one. So there are no true differential equations in SPICE.

That said, SPICE could output the transfer function of the circuit; IIRC as a list of polynomial coefficients. This sometimes suffers heavily from rounding errors but for simple circuits it may be what you are looking for.

I'll just comment on the design:

1. Replace C5 with a 100nF ceramic capacitor and place it close to the power supply pin of the MCP6022.
2. Put the designators on the PCB-Design, not values. Make it far easier to understand the layout.
3. Avoid 90° trace bends, they can cause problems when etching the board. They're also bad for high-speed stuff (at least that's the common opinion on the matter). Use two 45° bends instead.
4. Consider flooding one side of the board with a GND-Plane.
5. Use wide short traces for power supply connections.
6. Use one side of the board for mostly vertical traces and the other side for horizontal traces.
7. Take more care of component placement. Place them in a way where they are easier to route. Component placement is 70% of the job. Place them BEFORE starting to route a single trace (Won't always work out). Just use the ratsnest (the lines which indicate connections which are not routed yet) as a rough guideline.
8. Do not see a trace which is already routed as something which is fixed. If its in the way or you don't like they way it looks, rip it up and try again.
9. When in doubt, start from scratch, try not to rescue something which can't be rescued anymore.
10. Rule of thumb: Create something which pleases the eye. Others will have an easier time to understand it and sometimes it will even work better.

There are two Books i can highly recommend for learning Electronics/PCB Design: The Circuit Designer's Companion and EMC for Product Designers. While the second one more about EMC compliance it helps to understand WHY these things should be done in a certain way.