 Is the differential equation ordinary or partial? Partial differential equations describe multi-dimensional problems and are significantly more complex. Of which order is the differential equation? First order differential equations are usually easy to solve and describe, for example, exponential behavior such as radioactive decay or the cooling of a liquid. Differential equations of second order on the other hand are somewhat more complex and also often occur in nature. Maxwell's equations of electrodynamics, Schrodinger's equation of quantum mechanics, these are all second-order differential equations. Only starting from the second order a differential equation can describe in oscillation and only starting from the third order a differential equation can describe chaos. Is the differential equation linear or nonlinear? The superposition principle applies to linear differential equations which is incredibly useful, for example, in the description of electromagnetic phenomena. Nonlinear differential equations are much more complex and occur, for example, in nonlinear electronics in the description of superconducting currents. Moreover, chaos can only occur in nonlinear differential equations of third order and higher. When you encounter such an equation sometimes the only thing you can do is throw away your pen and paper and solve the equation numerically on the computer. Many nonlinear differential equations cannot even be solved analytically. Is the linear differential equation homogeneous or inhomogeneous? Homogeneous linear differential equations are simpler than inhomogeneous ones and describe, for example, an undisturbed oscillation while inhomogeneous differential equations are also able to describe externally disturbed oscillations. First, let's learn how to answer these questions. After you have classified a differential equation you can then specifically apply an appropriate method to solve the equation. Even if there is no specific solving method you will know how complex a differential equation is based on the classification. Our equation for the oscillating mass is an ordinary differential equation. Ordinary means that the function y we are looking for only depends on one variable, in this case on the time t. The wave equation on the other hand is a partial differential equation. Partial means that the searched function e depends on at least two variables and derivatives with respect to these variables occur in the equation. In this case, e depends on four variables t, x, y and z and in the differential equation also derivatives with respect to t, x, y, z appear. Furthermore, our equation for the oscillating mass is a differential equation of second order. The order of the differential equation is the highest occurring derivative of the searched function. Since in our equation the second derivative of y is the highest one, this is therefore the second order differential equation. The differential equation for the radioactive decay law on the other hand is a first order differential equation because the highest occurring derivative of the search function n is the first derivative. Moreover, our equation for the oscillating mass is linear. Linear means that the searched function and its derivatives contain only powers of one and there occur no products of derivatives with the function like y squared or y times the second derivative of y. There also occur no composed functions like sine of y or square root of y. Note that to the power of two in the second derivative in the Leibniz notation is not a power of the derivative but merely a notation that it is the second derivative. The radioactive decay law is also linear. What about the wave equation? It is also linear. The coupled differential equation system for the motion of a mass and the gravitational field on the other hand is non-linear. Here the searched functions x, y and z occur in quadratic form but even if the squares were not there, there would still be the square root and the fraction which make the differential equation system non-linear. In the next types of differential equations the coefficients multiplied by the searched function and its derivatives are important. In some solving methods it is important to distinguish between constant coefficients and non-constant coefficients. Constant coefficients do not depend on the variables on which the searched function depends. Non-constant coefficients do depend on the variables on which the searched function depends. A coefficient must not necessarily be multiplied with the searched function or its derivative. It can also stand alone. In this case we call the single coefficient a perturbation function. In our differential equation for the oscillating mass there is a constant coefficient which is multiplied by the searched function y namely d over m. Strictly speaking there is also a coefficient in front of the second derivative namely 1 and the single coefficient that is the perturbation function is 0 here so it does not exist. If the perturbation function is 0 then we call the linear differential equation homogeneous. So the differential equation for the oscillating mass is a homogeneous differential equation. The wave equation is also homogeneous because here there is also no single coefficient. The perturbation function is 0. The differential equation for a forced oscillation on the other hand is inhomogeneous. Here the external force f corresponds to the perturbation function. As you can see it stands alone without being multiplied by the function y or its derivatives. Moreover the perturbation function f is time dependent so it is a non-constant coefficient.