1. Solving Differential Equations

A differential equation (or "DE") contains derivatives or differentials.

Recall from the Differential section in the Integration chapter, that a differential can be thought of as a derivative where "dy/dx" is not written in fraction form.

Examples of Differentials

On this page...

Definitions of order & degree
General & particular solutions
Second order DEs

dx (this means "an infinitely small change in x")

dθ (this means "an infinitely small change in θ")

dt (this means "an infinitely small change in t")

Examples of Differential Equations

Example 1.

We saw the following example in the Introduction to this chapter. It involves a derivative (dy/dx):

As we did before, we would integrate it to produce the required solution. This will be a general solution (involving K, a constant of integration).

So we proceed as follows:

and this gives

In this example, we appear to be integrating the x part only, but in fact we have integrated with respect to y as well (on the left). Differential equations are like that - you need to integrate with respect to two different variables.

We could have written our question only using differentials:

dy = (x² − 3)dx

What we have done is to multiply both sides by dx.

Now we integrate both sides:

∫ dy = ∫(x² − 3)dx

Then the answer is the same as before:

On the left hand side, we have integrated ∫ dy = ∫ 1 dy to give us y.

Example 2

This example also involves differentials:

θ²dθ = sin(t + 0.2) dt

We have:

A function of θ with dθ on the left side, and

A function of t with dt on the right side.

To solve this, we would integrate both sides, as follows:

∫ θ²dθ = ∫ sin(t + 0.2) dt

θ³/3 = -cos(t + 0.2) + K

Solving a differential equation

From the above examples, we can see that solving a differential equation means finding an equation with no derivatives that satisfies the given differential equation. Solving a differential equation always involves one or more integration steps.

It is important to be able to identify the type of differential equation we are dealing with before we attempt to solve it.

Definitions

First order DE: Contains only first derivatives

Second order DE: Contains second derivatives (and possibly first derivatives also)

Degree: The highest power of the highest derivative which occurs in the differential equation.

Examples - Order and Degree

1) math formula

This DE has order 2 (the highest derivative appearing is the second derivative) and degree 1 (the power of the highest derivative is 1.)

2) math formula

This DE has order 1 (the highest derivative appearing is the first derivative) and degree 5 (the power of the highest derivative is 5.)

3) (y")⁴ + 2(y')⁷ − 5y = 3

This DE has order 2 (the highest derivative appearing is the second derivative) and degree 4 (the power of the highest derivative is 4.)

General and Particular Solutions

When we first performed integrations, we obtained a general solution (involving a constant, K).

We obtained a particular solution by substituting known values for x and y. These known conditions are called boundary conditions (or initial conditions).

It is the same concept when solving differential equations - find general solution first, then substitute given numbers to find particular solutions.

Let's see some examples of finding solutions of first order, first degree DEs.

Example 1

a. Find the general solution for the differential equation

dy + 7x dx = 0.

b. Find the particular solution given that y(0) = 3.

Answer

Example 2

Find the particular solution of

y' = 5

given that when x = 0, y = 2.

Answer

Example 3

Find the particular solution of

y''' = 0

given that:

y(0) = 3, y'(1) = 4, y''(2) = 6

Answer

Example 4

After solving the differential equation,

(we will see how to solve this DE in the next section Separation of Variables), we obtain the result

y = c ln x

Did we get the correct general solution?

Answer

Second Order DEs

We include two more examples here to give you an idea of second order DEs. We will see later in this chapter how to solve such Second Order Linear DEs.