Solving Differential Equations

A differential equation is an equation which contains a derivative such as or . Solving a differential equation means finding an expression for y in terms of x or for A in terms of t without the derivative. e.g. (1) e.g. (2) To solve (1) we just integrate with respect to x. However, we can’t integrate y w.r.t. x so (2) needs another method.

We can sketch the graph by drawing a gradient diagram. Before we see how to solve the equation, it’s useful to get some idea of the solution. The equation tells us that the graph of y has a gradient that always equals y. We can sketch the graph by drawing a gradient diagram. For example, at every point where y = 2, the gradient equals 2. We can draw a set of small lines showing this gradient. 2 1 We can cover the page with similar lines.

We can now draw a curve through any point following the gradients.

However, we haven’t got just one curve.

Can you guess what sort of equation these curves represent ? The solution is a family of curves. Can you guess what sort of equation these curves represent ? ANS: They are exponential curves.

Solving We use a method called “ Separating the Variables” and the title describes exactly what we do. We rearrange so that x terms are on the right and y on the left. Multiply by dx and divide by y. We can separate the 2 parts of the derivative because although it isn’t actually a fraction, it behaves like one. Now insert integration signs . . . and integrate (the l.h.s. is integrated w.r.t. y and the r.h.s. w.r.t. x) We don’t need a constant on both sides as they can be combined. I usually put it on the r.h.s.

We’ve now solved the differential equation to find the general solution but we have an implicit equation and we often want it to be explicit ( in the form y = . . . ) A log is just an index, so ( We now have the exponential that we spotted from the gradient diagram. ) However, it can be simplified.

We can write as . Since is a constant it can be replaced by a single letter, k. So, where k is positive This is usually written as where A is positive or negative. So, In this type of example, because the result is valid for positive and negative values, I usually use A directly when I change from log to exponential form.

Changing the value of A gives the different curves we saw on the gradient diagram. e.g. A = 2 gives

The differential equation is important as it is one of a group used to model actual situations. These are situations where there is exponential growth or decay. We will investigate them further in the next presentation. We will now solve some other equations using the method of separating the variables.

e.g. 3 Solve the equation below giving the answer in the form Solution: Separating the variables: Insert integration signs: Integrate:

e.g. 4 Solve the equation below giving the answer in the form Solution: It’s no good dividing by y as this would give which is no help. Instead, we take out x as a common factor on the r.h.s., so We can now separate the variables by dividing by

You may sometimes see this written as

SUMMARY Some differential equations can be solved by separating the variables. To use the method we need to be able to write the equation in the form ( If the equation has a total of 3 terms we will need to bracket 2 together before separating the variables. ) The l.h.s. is integrated w.r.t. y and the r.h.s. w.r.t. x, so The answer is often written explicitly. The solution is called the general solution.

Exercise 1. Find the general solutions of the following equations giving your answers in the form : (b) (a) 2. Find the general solutions of the following equations leaving the answers in implicit form: (a) b) 3. Find the equation of the curve given by the following equation and which passes through the given point. ( This is called a particular solution. )

Solutions: (b)

2(a) b)

3. You might prefer to write as before you separate the variables. or

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

e.g. 1 Solving We use a method called “ Separating the Variables” and the title describes exactly what we do. We rearrange so that x terms are on the right and y on the left. Now insert integration signs . . . and integrate (the l.h.s. is integrated w.r.t. y and the r.h.s. w.r.t. x) Multiply by dx and divide by y.

We’ve now solved the differential equation to find the general solution but we have an implicit equation and we often want it to be explicit ( in the form y = . . . ) A log is just an index, so We usually use A. So, We can write as . Since is a constant it can be replaced by a single letter. This can be simplified.

e.g. 2 Solve the equation below giving the answer in the form Solution: Separating the variables: Insert integration signs: Integrate:

e.g. 3 Solve the equation below giving the answer in the form Solution: It’s no good dividing by y as this would give which is no help. Instead, we take out x as a common factor on the r.h.s. We can now separate the variables by dividing by

SUMMARY Some differential equations can be solved by separating the variables. To use the method we need to be able to write the equation in the form ( If the equation has a total of 3 terms we will need to bracket 2 together before separating the variables. ) The l.h.s. is integrated w.r.t. y and the r.h.s. w.r.t. x, so The answer is often written explicitly. The solution is called the general solution.