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# Numerical Solutions of Ordinary Differential Equations
Fundamental laws of the universe derived in fields such as, physics, mechanics, electricity and thermodynamics define mechanisms of change. When combining these law's with continuity laws of energy, mass and momentum we get differential equations.
| Newton's Second Law of Motion | Equation | Description |
| ----------------------------- | -------------------------------- | ---------------------------------------------------- |
| Newton's Second Law of Motion | $$\frac{dv}{dt}=\frac{F}{m}$$ | Motion |
| Fourier's heat law | $$q=-kA\frac{dT}{dx}$$ | How heat is conducted through a material |
| Fick's law of diffusion | $$J=-D\frac{dc}{dx}$$ | Movement of particles from high to low concentration |
| Faraday's law | $$\Delta V_L = L \frac{di}{dt}$$ | Voltage drop across an inductor |
In engineering ordinary differential equation's (ODE) are very common in the thermo-fluid science's, mechanics and control systems. By now you've solve many ODE's however probably not using numerical methods. Suppose we have an initial value problem of the form
$$
\frac{dy}{dt}=f(t,y), \quad y(t_0)=y_0
$$
where $f(t,y)$ describes the rate of change of $y$ with respect to time $t$.
[[3_6_Explicit_Methods]]
[[3_7_Implicit_Method]]
[[3_8_Systems_of_ODEs]]
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