FP2 Chapter 4 – First Order Differential Equations

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 6th August 2015

Intro

Differential equations are equations which relate and with derivatives. e.g.

The rate of temperature loss is proportional to the current temperature.

𝑑𝑇𝑑𝑡

=−𝑘𝑇

The rate of population change is proportional to where is the current population and is the limiting size of the population (the Verhulst-Pearl Model)

𝑑𝑃𝑑𝑡

=𝑘𝑃(1− 𝑃𝑀 )

As you might imagine, they’re used a lot in physics and engineering, including modelling radioactive decay, mixing fluids, cooling materials and bodies falling under gravity against resistance.

A ‘first order’ differential equation means the equation contains the first derivative () but not the second derivative or beyond.

Suppose is GDP (Gross Domestic Product).Rate of change of GDP is proportional to current GDP.

𝑑𝑥𝑑𝑡

=𝑘𝑥

C4 recap

and are said to be ‘separated’ because they are not mixed in the same expression.

Divide through by and times through by .

Slap and integral symbol on the front!

Examples

Find general solutions to

𝑦=2 𝑥+𝐶Why is it called the general solution?We have a ‘family’ of solutions as the constant of integration varies.?

Find general solutions to

If we let , we get

𝑥

𝑦So the ‘family of circles’ satisfies this differential equation.

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Examples

Find general solutions to

where

Find general solutions to

(a family of rectangular hyperbolae)

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Particular solutionsJust like at the end of C1 integration, we can fix to one particular solution if we give some conditions.

Find the particular solutions for which satisfy the initial conditions when .

∫𝑥− 12 𝑑𝑥=∫ 1𝑑𝑦

Some people (including myself) like to sub in as soon as possible before rearranging.?

Exercise 4A

Using reverse product ruleWe will see in a bit how to solve equations of the form (where and are functions of ). We’ll practice a particular part of this method before going for the full whack.

Find general solutions of the equation

We can’t separate the variables. But do you notice anything about the LHS?It’s !

Quickfire Questions:

𝑑𝑑𝑥

(𝑥2 𝑦 )=𝒙𝟐 𝒅𝒚𝒅𝒙

+𝟐 𝒙𝒚

So it appears whatever term ends up on front of the will be on the front of the in the integral.

𝑥4𝑑𝑦𝑑𝑥

+4 𝑥3 𝑦→𝒅𝒅𝒙

(𝒙𝟒 𝒚 )?

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??

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Using reverse product rule

Find general solutions of the equation

𝑑𝑑𝑥

(𝑥3 𝑦 )=sin 𝑥Test Your Understanding

Find general solutions of the equation Find general solutions of the equation

𝑑𝑑𝑥 ( 𝑦𝑥 )=𝑒𝑥 𝑑

𝑑𝑥(2𝑥 𝑦2 )=𝑥2

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? ?

Find the general solution of

But what if we can’t use the product rule backwards?

We can multiply through by the integrating factor . This then produces an equation where we can use the previous reverse-product-rule trick (we’ll prove this in a sec).

𝐼 .𝐹 .=𝑒∫− 4𝑑𝑥=𝑒−4 𝑥

𝑒−4 𝑥𝑑𝑦𝑑𝑥−4 𝑒− 4 𝑥𝑦=𝑒−3 𝑥

Then multiplying through by the integrating factor:

Then we can solve in the usual way:

𝑑𝑑𝑥

( 𝑦 𝑒−4 𝑥 )=𝑒− 3𝑥

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Proof that Integrating Factor works

Solve the general equation , where are functions of .

Suppose is the Integrating Factor. As usual we’d multiply by it:

If we can use the reverse product rule trick on the LHS, then it would be of the form:

Thus comparing the coefficients of the two LHSs:

Dividing by and integrating:

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You could skip to here provided you don’t forget to multiply the RHS by the IF.

When there’s something on front of the

Find the general solution of

What shall we do first so that we have an equation like before?

𝑑𝑦𝑑𝑥

+2 𝑦 tan 𝑥=cos3𝑥STEP 1: Divide by anything on front of

STEP 2: Determine IF

STEP 3: Multiply through by IF and use product rule backwards.

STEP 4: Integrate and simplify.

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Test Your Understanding

Find the general solution of the differential equationFP2 June 2011 Q3

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Exercise 4C

Making a substitution

Sometimes making a substitution will turn a more complex differential equation into one like we’ve seen before.

Use the substitution to transform the differential equation into a differential equation in and . By first solving this new equation, find the general solution of the original equation, giving in terms of .

𝑧=𝑦𝑥→ 𝑦=𝑥𝑧

Make the subject so that we can find and sub out.

Sub in and

Simplify to either get or

Sub back in at the end.

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Another with an IF

Use the substitution to transform the differential equation , into a differential equation in and . Find the general solution using an integrating factor.

𝑦=1𝑧,𝑑𝑦𝑑𝑥

=−1

𝑧2𝑑𝑧𝑑𝑥

The and are separable here (but factorising out ), so we could have done without an IF.?

Test Your Understanding

(a) Show that the substitution transforms the differential equation

into the differential equation

(b) By solving differential equation (II), find a general solution of differential equation (I) in the form .(c) Given that at , find the value of at .

FP2 June 2012 Q7

a ?

b ?

c ?

Exercise 4D/E