# Examples of differential equations

**Differential equations** arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in a few simple cases when an exact solution exists.

## Contents

- 1 Separable first-order ordinary differential equations
- 2 Separable (homogeneous) first-order linear ordinary differential equations
- 3 Non-separable (non-homogeneous) first-order linear ordinary differential equations
- 4 Second-order linear ordinary differential equations
- 5 Linear systems of ODEs
- 6 See also
- 7 Bibliography
- 8 External links

## Separable first-order ordinary differential equations[edit]

Equations in the form are called separable and solved by and thus . Prior to dividing by , one needs to check if there are stationary (also called equilibrium) solutions satisfying .

## Separable (homogeneous) first-order linear ordinary differential equations[edit]

A separable *linear* ordinary differential equation of the first order
must be homogeneous and has the general form

where is some known function. We may solve this by separation of variables (moving the *y* terms to one side and the *t* terms to the other side),

Since the separation of variables in this case involves dividing by *y*, we must check if the constant function *y=0* is a solution of the original equation. Trivially, if *y=0* then *y'=0*, so *y=0* is actually a solution of the original equation. We note that *y=0* is not allowed in the transformed equation.

We solve the transformed equation with the variables already separated by Integrating,

where *C* is an arbitrary constant. Then, by exponentiation, we obtain

- .

Here, , so . But we have independently checked that *y=0* is also a solution of the original equation, thus

- .

with an arbitrary constant *A*, which covers all the cases. It is easy to confirm that this is a solution by plugging it into the original differential equation:

Some elaboration is needed because *ƒ*(*t*) might not even be integrable. One must also assume something about the domains of the functions involved before the equation is fully defined; the solution above assumes the real case.

If is a constant, the solution is particularly simple, and describes, e.g., if , the exponential decay of radioactive material at the macroscopic level. If the value of is not known a priori, it can be determined from two measurements of the solution. For example,

gives and .

## Non-separable (non-homogeneous) first-order linear ordinary differential equations[edit]

First-order linear non-homogeneous ODEs (ordinary differential equations) are not separable, they can be solved by the following approach, known as an *integrating factor* method. Consider first-order linear ODEs of the general form:

The method for solving this equation relies on a special integrating factor, *μ*:

We choose this integrating factor because it has the special property that its derivative is itself times the function we are integrating, that is:

Multiply both sides of the original differential equation by *μ* to get:

Because of the special *μ* we picked, we may substitute *dμ*/*dx* for *μ* *p*(*x*), simplifying the equation to:

Using the product rule in reverse, we get:

Integrating both sides:

Finally, to solve for *y* we divide both sides by :

Since *μ* is a function of *x*, we cannot simplify any further directly.

## Second-order linear ordinary differential equations[edit]

### A simple example[edit]

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension/compression of the spring. For now, we may ignore any other forces (gravity, friction, etc.). We shall write the extension of the spring at a time *t* as *x*(*t*). Now, using Newton's second law we can write (using convenient units):

where *m* is the mass and *k* is the spring constant that represents a measure of spring stiffness. For simplicity's sake, let us take *m=k* as an example.

If we look for solutions that have the form , where *C* is a constant, we discover the relationship , and thus must be one of the complex numbers or . Thus, using Euler's formula we can say that the solution must be of the form:

See a solution by WolframAlpha.

To determine the unknown constants *A* and *B*, we need *initial conditions*, i.e. equalities that specify the state of the system at a given time (usually *t* = 0).

For example, if we suppose at *t* = 0 the extension is a unit distance (*x* = 1), and the particle is not moving (*dx*/*dt* = 0). We have

and so *A* = 1.

and so *B* = 0.

Therefore *x*(*t*) = cos *t*; this is an example of simple harmonic motion.

See a solution by Wolfram Alpha.

### A more complicated model[edit]

The above model of an oscillating mass on a spring is plausible but not very realistic: in practice, friction will tend to decelerate the mass and have magnitude proportional to its velocity (i.e. *dx*/*dt*). Our new differential equation, expressing the balancing of the acceleration and the forces, is

where is the damping coefficient representing friction. Again looking for solutions of the form , we find that

This is a quadratic equation which we can solve. If there are two complex conjugate roots *a* ± *ib*, and the solution (with the above boundary conditions) will look like this:

Let us for simplicity take , then and .

The equation can be also solved in MATLAB symbolic toolbox as

```
x = dsolve('D2x+c*Dx+k*x=0','x(0)=1','Dx(0)=0')
```

although the solution looks rather ugly,

```
x = (c + (c^2 - 4*k)^(1/2))/(2*exp(t*(c/2 - (c^2 - 4*k)^(1/2)/2))*(c^2 - 4*k)^(1/2)) -
(c - (c^2 - 4*k)^(1/2))/(2*exp(t*(c/2 + (c^2 - 4*k)^(1/2)/2))*(c^2 - 4*k)^(1/2))
```

This is a model of a damped oscillator; the plot of displacement against time would look like this:

which resembles how one would expect a vibrating spring to behave as friction removes energy from the system.

## Linear systems of ODEs[edit]

The following example of a first order linear systems of ODEs

can be easily solved symbolically using numerical analysis software.

## See also[edit]

## Bibliography[edit]

- A. D. Polyanin and V. F. Zaitsev,
*Handbook of Exact Solutions for Ordinary Differential Equations*, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003; ISBN 1-58488-297-2.

## External links[edit]

- Ordinary Differential Equations at EqWorld: The World of Mathematical Equations.