Introduction to Differential Equations

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

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

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

So, the above equations can be rewritten with another notation as follows: 1td2ydt2+dydt+ty=0\frac{{1}}{{t}}\frac{{{{d}}^{{2}}{y}}}{{{d}{{t}}^{{2}}}}+\frac{{{d}{y}}}{{{d}{t}}}+{t}{y}={0} and (d3ydt3)4+d2ydt2dydt=5t{{\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 (ut)2+ux=x+t{{\left(\frac{{\partial{u}}}{{\partial{t}}}\right)}}^{{2}}+\frac{{\partial{u}}}{{\partial{x}}}={x}+{t} and ut+ux=(uv)2\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, yty=y{y}''-{t}{y}={y}''' is a third-order differential equation, while yty=2{y}'-{t}{y}={2} is a first-order one.

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

For example, y(t)=et{y}{\left({t}\right)}={{e}}^{{t}} is a solution to the differential equation yy=0{y}'-{y}={0}, because y=et{y}'={{e}}^{{t}} and etet=0{{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+3y+2y=0{y}''+{3}{y}'+{2}{y}={0}, y(1)=1{y}{\left({1}\right)}={1}, y(1)=3{y}'{\left({1}\right)}={3} is an IVP, and the conditions y(1)=1{y}{\left({1}\right)}={1}, y(1)=3{y}'{\left({1}\right)}={3} are initial. Another example of initial conditions for the above problem is y(0)=0{y}{\left({0}\right)}={0}, y(0)=3{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+3y+2y=0{y}''+{3}{y}'+{2}{y}={0}, y(1)=1{y}{\left({1}\right)}={1}, y(2)=3{y}{\left({2}\right)}={3} is a BVP, and the conditions y(1)=1{y}{\left({1}\right)}={1}, y(2)=3{y}{\left({2}\right)}={3} are boundary. Another example of boundary conditions for the above problem is y(0)=0{y}{\left({0}\right)}={0}, y(4)=5{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(t){y}={y}{\left({t}\right)}, or, in other words, if the left-hand side equals y{y} and the right-hand side doesn't contain y{y}.

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

A solution is called implicit, if we can't express y{y} in terms of independent variables. For example, y3+ln(ty)=0{{y}}^{{3}}+{\ln{{\left({t}{y}\right)}}}={0} and yey=t\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 (y)2=1{{\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=y{y}'=\sqrt{{{y}}} has at least 2 solutions: y=14t2{y}=\frac{{1}}{{4}}{{t}}^{{{2}}} and y=0{y}={0}.
  • If the differential equation has a solution, is it possible to find it? The answer is not always yes.