Difference between ordinary and partial differential equations

Question

Being an undergraduate student I find difficult to understand the perfect differences between normal and partial differential equations. Elaborate the answer

Answer 1

Let's look at some examples of ODEs and PDEs in physics:

1) A particle moves under the influence of gravity, electromagnetic forces, viscosity or other forces. The position of the particle is a function of a single independent variable (time) so we can represent the equation of motion of the particle by an ODE.

2) A chain hangs under its own weight, and has static loads attached to it at fixed points. The deflection of the chain is a function of a single independent variable (the distance along the chain) so again the deflection of the chain satisfies an ODE.

3) A string that is fixed at both ends is displaced from its equilibrium position and released. The deflection of the string is now a function of two independent variables - time and distance along the string - so its equation of motion must be represented by a PDE, which is the "wave equation".

4) An insulated metal bar starts at a uniform temperature. Its ends are then heated to two different constant temperatures. Initially the temperature of the bar is a function of two independent variables - time and distance along the bar. So the temperature of the bar satisfies a PDE - the heat equation. As time goes by, the temperature of the bar approaches an equilibrium state where its temperature does not depend on time, but only on the distance along the bar. The equilibrium temperature (as a function of distance only) can be found by solving an ODE.

Answer 2

An ordinary differential equation involves a derivative over a single variable, usually in an univariate context, whereas a partial differential equation involves several (partial) derivatives over several variables, in a multivariate context.

E.g. $$\frac{dz(x)}{dx}=z(x)$$

vs.

$$\frac{\partial z(x,y)}{\partial x}+\frac{\partial z(x,y)}{\partial y}=z(x,y).$$

PDEs are notably more difficult to solve than ODEs.

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Remark:

Even though several variables appear,

$$\frac{\partial z(x,y)}{\partial x}=z(x,y)$$

can very well be considered as an ODE, where $y$ is just an independent parameter.