# 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.