1

First Order Linear Differential Equations

Chapter 4 FP2 EdExcel

2

Splitting the Variables This is a method used in C4 of the Core

Mathematics A level and then in Chapter 4 FP2. When:

We can split the variable in order to find a final equation in the form : y=f(x)

We do this by Dividing and Multiplying to get g(y) on one side and f(x) on the other side, and splitting dy/dx up to the appropriate sides like so:

We can now integrate to find y in the form f(x). Do not forget the arbitrary constant when doing so, and if given values for x and y, use them to find the value of this constant.

• Worked example (Elmwood C4, paper A)

)()( ygxfdx

dy

dxxfdyyg

)()(

1

Cx

xy

Cx

x

y

Cx

xy

dxxxy

dxxx

xdy

y

dxx

xdy

y

xydx

dyx

1

11

11

41

141

141

)14(

5

5

4

23

22

5

2

2

5

2

522

3

dy/dx +p(x)y=q(x)Multiplying Factors

• The problem in such a differential equation is that one can not simply divide and multiply to split the variables. We therefore need to manipulate the expression to allow calculus to be used to solve the problem.

• In order to do this we need to think back to C3 calculus and remember the product rule.

• We need to manipulate the equation into this form, and we can do this by using a multiplying factor, M(x).

• By doing this, we have allowed ourselves to manipulate the equation into the form of the product rule, because we can let M’(x)=M(x)q(x), and thus find M(x).

• We know that:

• This gives us a generalised form of our Multiplying factor M(x). We can ignore arbitrary constants in finding M(x) as we only need a solution, not a particular one, and they would only cancel in the equation anyway.

)(uvdx

d

dx

dvu

dx

duv

)()()()()( xqxMyxpxMxM dxdy

dxxpexM

dxxpxM

dxxpdxxM

xM

xpxM

xM

xpxMxM

)()(

)()(ln

)()(

)(

)()(

)(

)()()(

4

Using the Multiplying factors• We know that:

• We also know that:

• We now get the form:

• The LHS takes the form of the product rule and can be changed to:

• So we get:

• Which can be solved using calculus.

• Worked example (EdExcel FP1 Jan ’08):

dxxp

exM)(

)(

)()()()()( xqxMyxpxMdx

dyxM

)()()()( xqxMyxMdx

dyxM

)(. xMydx

d

)()()(. xqxMxMydx

d

x

xxx

xx

xxx

x

dxxp

Cex

y

partsbyCexe

ye

xeyedx

d

xeyedx

dye

exM

exM

xxMyxMdx

dyxM

xydx

dy

3

333

33

333

3

)(

9

1

3

...........93

.

).(

3

)(

)(

)(3)()(

3

5

Substitutions• One can use substitutions in order to transform

differential equations into more easily solvable problems. These substitutions eliminate y from the equation and create a differential equation in terms of x and z (where z is the substitution).

» This can now be solved by splitting the variables as shown on page 2.

• One can also use substitutions in order to eliminate x and y and create a simple calculus problem to solve.

» This can now be solved by splitting the variables as shown on page 2.

z

z

dx

dzx

zz

z

dx

dzx

zx

zxz

dx

dzx

zdx

dzx

dx

dy

xzyx

yzlet

xy

yx

dx

dy

2

1

2

)31(

2

)31(

_

2

3

2

2

2

22

22

3

13

32

13

23

21

1

_

3

2

udx

duu

uu

dx

duu

u

dx

duu

u

dx

dudx

dy

dx

du

xyulet

xy

xy

dx

dy