Ordinary Differential Equations/Introduction

Directly integrable differential equationsEdit

An example of an ordinary differential equation is the equation


To solve this equation shall mean that we find a function   defined on some interval   such that the above equation holds for all  . That is, as opposed to "normal" algebraic equations such as


to which the solution is a number (in this case  ), the solution to an ordinary differential equation is a function.

There is a theorem in calculus that says that a differentiable function   on a connected interval   has vanishing derivative if and only if it is a constant function. Thus, the solutions to the equation


are precisely the constant functions   ( ). Note here that we have lost the uniqueness of solutions (which we had in our example for an algebraic equation); every constant function is a solution. We will later be able to restore the uniqueness of solutions (at least in some special cases) by imposing initial conditions; for example, if we additionally require that


(where we also require that   is contained within the solution interval  ), then we have only one choice for a solution to our equation:  .

Let's turn to a slightly more difficult differential equation:


for some function  . In this case, by the fundamental theorem of calculus we have

  for some   to be contained in the solution interval  ,

where again   is an arbitrary integration constant. The arbitraryness of   of course implies that we have infinitely many solutions, but by imposing an initial condition  , we get that   must equal  , and hence we have once again obtained a unique solution if we additionally require the initial condition to be satisfied.


  • Solve the differential equation   under the initial condition  . More generally, solve the differential equation   under the initial condition  .

Order of an ordinary differential equationEdit

Instead of looking at the equation


we may instead look at the equation

 , or, more generally,  

for some function  . Note the difference: Previously, we had only had a first-order derivative, and now we have a second order derivative. The new equation is still an ordinary differential equation (we will give the precise definition of an ordinary differential equation at the end, when we are finished with the simple examples), but this time involving a second-order derivative. This gives rise to the following definition.

Definition 1.1 (order of an ordinary differential equation):

The order of an ordinary differential equation is defined to be the order of the highest order derivative occuring in it. That is, if   is appearing in the ordinary differential equation, but for all  ,   does not appear in the ordinary differential equation, then   is the order of that ordinary differential equation.

In fact, for the above examples, we can easily compute the solutions using just integration. If  , then   for some constant  . Therefore, integrating that again, we get


for some constant  .


  • What is the order of the ordinary differential equation  ? What is the order of the ODE  ? Based on the last question, write down an ordinary differential equation of order 23.
  • Solve the differential equation   under the conditions   and  .

Linear differential equationsEdit

Consider the differential equation


where   is an arbitrary constant. This differential equation has a remarkable property: Whenever   are solutions to it, so is   for arbitrary   (in fact, complex numbers are permissible as well). More generally, every linear combination of solutions is again a solution. This is a direct consequence of linearity of the derivatives as follows:


Differential equations with this property are called linear.

Definition 1.2 (linear ordinary differential equation):

A linear ordinary differential equation is an ODE such that any linear combination of solutions to it is again a solution to it.

Note that since zero can be written as a linear combination of any existing solution (as  ), a linear ODE has either no solution or zero is a solution.

There is also such a thing as an inhomogenous linear equation, which is closely related, but not equal, to linear equations.

Consider the ordinary differential equation


for some function  . This is an example of what's called an inhomogenous linear equation, for the following reason: Suppose it has a solution  , and   is a solution of the above equation  . Then any superposition   ( ) is again a solution:


We capture this property in the following definition.

Definition 1.3 (inhomogenous linear ordinary differential equation):

An inhomogenous linear ordinary differential equation is an ODE such that there is a corresponding linear ODE, of which we can add solutions and obtain still a solution.

To contrast linear ODEs from inhomogeneous linear ODEs, we often call the former homogenous linear ODEs. Some mathematicians also use the term "linear" to refer to homogenous or inhomogenous ODEs, which is why it is advisable to use the term "homogenous" anyway.

We will see only later how to solve these things, although it's actually very easy.


Let  . Prove by direct computation that the function   is a solution to the ODE  .

More than one dimensionEdit

Autonomous differential equationsEdit

Delimitation from partial differential equationsEdit

The precise definition of an ordinary differential equationEdit