Introduction to Differential Equations

To put it simply, a differential equation is any equation that contains derivatives.

For example, $\frac{{{y}''}}{{t}}+{y}'+{t}{y}={0}$ and ${{\left({y}'''\right)}}^{{4}}+\sqrt{{{y}''}}-{y}'={5}{t}$ are both differential equations.

Different notations can be used: either ${{y}}^{{{\left({n}\right)}}}$ or $\frac{{{{d}}^{{n}}{y}}}{{{d}{{t}}^{{n}}}}$ .

So, the above equations can be rewritten with another notation as follows: $\frac{{1}}{{t}}\frac{{{{d}}^{{2}}{y}}}{{{d}{{t}}^{{2}}}}+\frac{{{d}{y}}}{{{d}{t}}}+{t}{y}={0}$ and ${{\left(\frac{{{{d}}^{{3}}{y}}}{{{d}{{t}}^{{3}}}}\right)}}^{{4}}+\sqrt{{\frac{{{{d}}^{{2}}{y}}}{{{d}{{t}}^{{2}}}}}}-\frac{{{d}{y}}}{{{d}{t}}}={5}{t}$ .

A differential equation is ordinary, if it contains derivatives with respect to only one variable.

A differential equation is called partial, if it contains derivatives with respect to more than one variable.

So, the above equations are examples of ordinar differential equations (ODE), and both ${{\left(\frac{{\partial{u}}}{{\partial{t}}}\right)}}^{{2}}+\frac{{\partial{u}}}{{\partial{x}}}={x}+{t}$ and $\frac{{\partial{u}}}{{\partial{t}}}+\frac{{\partial{u}}}{{\partial{x}}}={{\left(\frac{{\partial{u}}}{{\partial{v}}}\right)}}^{{2}}$ are examples of partial differential equations (PDE).

The order of a differential equation is the order of the highest derivative in the equation.

For example, ${y}''-{t}{y}={y}'''$ is a third-order differential equation, while ${y}'-{t}{y}={2}$ is a first-order one.

A solution of a differential equation is an unknown function ( ${y}$ or ${u}$ ) along with the independent variables that satisfy the given differential equation.

For example, ${y}{\left({t}\right)}={{e}}^{{t}}$ is a solution to the differential equation ${y}'-{y}={0}$ , because ${y}'={{e}}^{{t}}$ and ${{e}}^{{t}}-{{e}}^{{t}}={0}$ .

In general, each differential equation has a set of solutions; to restrict it to one particular solution, supplementary conditions are added. To find a particular solution of an nth-order differential equation, n conditions have to be added.

Conditions are called initial, if the values of a function and its derivatives are given at the same value of the independent variable, and the corresponding problem is called the initial value problem (IVP).

For example, ${y}''+{3}{y}'+{2}{y}={0}$ , ${y}{\left({1}\right)}={1}$ , ${y}'{\left({1}\right)}={3}$ is an IVP, and the conditions ${y}{\left({1}\right)}={1}$ , ${y}'{\left({1}\right)}={3}$ are initial. Another example of initial conditions for the above problem is ${y}{\left({0}\right)}={0}$ , ${y}'{\left({0}\right)}={3}$ .

Conditions are called boundary, if they are given at more than one value of the independent variable, and the corresponding problem is called the boundary value problem (BVP).

For example, ${y}''+{3}{y}'+{2}{y}={0}$ , ${y}{\left({1}\right)}={1}$ , ${y}{\left({2}\right)}={3}$ is a BVP, and the conditions ${y}{\left({1}\right)}={1}$ , ${y}{\left({2}\right)}={3}$ are boundary. Another example of boundary conditions for the above problem is ${y}{\left({0}\right)}={0}$ , ${y}{\left({4}\right)}={5}$ .

So, a particular solution is a solution that satisfies the differential equation and the given conditions. A general solution is a solution in the most general form that only satisfies the differential equation without satisfying any additional conditions.

A solution is called explicit, if ${y}={y}{\left({t}\right)}$ , or, in other words, if the left-hand side equals ${y}$ and the right-hand side doesn't contain ${y}$ .

For example, ${y}={{e}}^{{t}}+\frac{{\ln{{\left({t}\right)}}}}{{t}}$ and ${y}={\ln{{\left(\sqrt{{{t}}}\right)}}}+{t}$ are explicit solutions.

A solution is called implicit, if we can't express ${y}$ in terms of independent variables. For example, ${{y}}^{{3}}+{\ln{{\left({t}{y}\right)}}}={0}$ and $\sqrt{{{y}{{e}}^{{y}}}}={t}$ are implicit solutions.

Before solving a differential equation, it is essential to ask yourself the following 3 questions:

Does the given differential equation have a solution? Not all differential equations have solutions, for example ${{\left({y}'\right)}}^{{2}}=-{1}$ doesn't have any solutions, because the left side is never negative and the right side is negative.
If the differential equation has solutions, how many solutions are there? For example, ${y}'=\sqrt{{{y}}}$ has at least 2 solutions: ${y}=\frac{{1}}{{4}}{{t}}^{{{2}}}$ and ${y}={0}$ .
If the differential equation has a solution, is it possible to find it? The answer is not always yes.