SPECIALIST MATHS

Differential Equations

Week 1

Differential Equations

• The solution to a differential equations is a function that obeys it.

• Types of equations we will study are of the form:

)(dx

dy (1) xf

)( (2) ygdx

dy

)()( (3) ygxfdx

dy

Obtaining Differential Equations

• To obtain a differential equation from a function, we must:

• differentiate the function, then

• manipulate the result to achieve the appropriate equation.

Example 1 (Ex 8B1)•Show that is a solution of

the differential equation

12ln53 3 xxy

12

109 2

xx

dx

dy

Solution 1• Show that is a solution of

the differential equation

12ln53 3 xxy

12

109 2

xx

dx

dy

12ln53 3 xxy

212

1533 2

xx

dx

dy

12

109 2

xx

dx

dy

Example 2 (Ex 8B1)Show that is a solution of the differential equation

1tan xxy

xdx

dy 2tan

Solution 2Show that is a solution of

the differential equation

Solution

1tan xxy

xdx

dy 2tan

1tan xxy

1sec2 xdx

dy

11tan2 xdx

dy

xdx

dy 2tan

AA 22 sec1tan

Example 3 (Ex 8B1)Show that is a solution of the differential equation

BAey kx

)( Bykdx

dy

Solution 3Show that is a solution of

the differential equation

Solution

BAey kx

)( Bykdx

dy

BAey kx

kxkAedx

dy BAey kx

kxAeBy

Now

)( Bykdx

dy

Example 4 (Ex 8B2)Given is the solution of the differential equation

Find a, b, c and d given

dbxxaxy c ln)(38

2)(' x

xxy

4)1( y

Solution 4Given is the solution of

the differential equation

Find a, b, c and d given

Solution:

dbxxaxy c ln)(38

2)(' x

xxy

4)1( y

dbxxaxy c ln)(

1 )(' ccbxx

axy

2a31c

4c

382

)(' xx

xy

Solution 4 continueddbxxaxy c ln)(

2a 4c8cb84 b

2bdxxxy 4 2ln2)(

dy 4 )1(21ln2)1(

d 12024

d 246d

1 )(' ccbxx

axy

382

)(' xx

xy

,2a ,2b & 4c 6d

4)1( y

Example 5 (Ex 8B2)Find a, b, c, and d if is the solution of and

dcxbxaxy sin)(xxy 2sin12)('' 5)(' oy

7)( and y

Solution 5Find a, b, c, and d if is

the solution of and

Solution:

dcxbxaxy sin)(xxy 2sin12)(''

5)(' oy 7)( and y

dcxbxaxy sin)(

cbxabxy cos)('

bxabxy sin)('' 2 xxy 2sin12)(''

2b 122 ab1222 a

3a124 a

Solution 5 continuedcbxabxy cos)(' 3a 2b

5)0(' ycxxy 2cos23)('

cy )02cos(23)0('

c )0cos(65c 165

c 65

1cdcxbxaxy sin)(

dxxxy 2sin3)(

Solution 5 continued again

dxxxy 2sin3)(

7)( ydy 2sin3)(

d 037

d 7

d 7

3a 2b 1c 7d

Slope Fields

• The differential equation gives a formula for the slope its solutions.

• For example the differential equation gives an equation to calculate the slopes of all

points in the plane for functions whose derivatives are .

• That is it gives the slopes of all points of functions of the form

xdx

dy2

x2

cxy 2

Slope Field for f ‘(x) = 2x

x=0

x=1 x=2x=-1x=-2

y

x

Slope Field Generator

• y’ = 2x for y = x2 + c

• y‘ = 3x2 for y = x3 + c

• y’ = 2x + 1 for y = x2 + x + c

• y’ = x

• y’ = y

• y’ = x + y

http://alamos.math.arizona.edu/ODEApplet/JOdeApplet.html

Example 6 (Ex 8C1)Solve the following differential equation

xxy 2sin42)('

Solution 6Solve the following differential equation

Solution:xxy 2sin42)('

xxy 2sin42)('

dxxxy 2sin42)(

cxx 2cos2

142

cxx 2cos22

Example 7 (Ex 8C1)

Solve 10)1( and ,5

31

Pt

tdt

dP

t

Solution 7Solve

Solution

10)1( and ,5

31

Pt

tdt

dP

t

tt

dt

dP

t

53

1

ttt

dt

dP 53

532 ttdt

dP

Solution 7 continued

dtttP )53( 2

cP

)1(52

)1(3

3

)1()1(

23

10)1( P

c 52

3

3

110

c 52

3

3

110

cxxx

xP 52

3

3)(

23

Solution 7 continued again

6

515c

cxxx

xP 52

3

3)(

23

c 52

3

3

110

6

5155

2

3

3)(

23

xxx

xP

Euler’s Method of Numerical Integration

• We find the solution of a differential equation by moving small increments along the slope field

• Start at (xo,yo), then move up the slope field and at the same time going out horizontally h to get to the next point (x1,y1).

• The smaller the value of h the more accurate the solution.

Euler’s Methodhxx 01

),( 00 yx h

),( 11 yx

)(' 0xfm 01

01

xx

yym

0101 xxmyy

)(' 001 xhfyy

0101 yyxxm

)('y

GeneralIn

11n

1

nn

nn

xhfy

hxx

00001 )(' xhxxfyy

Fundamental Theorem of Calculus

• Using Euler’s method if we make the size of h very small then the y value of the point we approach is given by:

b

adxxyayby

)(')()(

Example 8 (Ex 8C2)Use Euler’s method with 3 steps to find y(0.6) for the differential equation with y(0)=2

Find y(6) using the Fundamental theorem

xdx

dysin

Solution 8Use Euler’s method with 3 steps to find y(0.6) for

the differential equation with y(0)=2

Find y(6) using the Fundamental theorem

Solution:

xdx

dysin

2 ,0 2)0( 00 yxy

2.0 ,6.0 to0 from steps 3 hxx

2.02.0001 hxx

2)0sin(2.02)(' 001 xhfyy

Solution 8 continued2.01 x 21 y

9603.1)2.0sin(2.02)(' 112 xhfyy

x 6.02.04.023 hxx

4.02.02.012 hxx

8824.1)4.0sin(2.09603.1)(' 223 xhfyy

8824.1)6.0( y

Solution 8 continued againTheorem lFundamenta theUsing

b

adxxyayby

)(')()(

0.6

0 )(')0()6.0( dxxyyy

0.6

0 )sin(2)6.0( dxxy

8253.1

This week

• Exercise 8A1 Q2, 3

• Exercise 8B1 Q 1 – 7

• Exercise 8B2 Q 1 – 7

• Exercise 8C1 Q 1 – 7

• Exercise 8C2 Q 1, 2