© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
47: Solving Differential 47: Solving Differential EquationsEquations
Solving Differential Equations
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Module C4
Solving Differential Equations
A differential equation is an equation which
contains a derivative such as or . dx
dy
dt
dA
e.g. (1) xdx
dy e.g. (2) y
dx
dy
Solving a differential equation means finding an expression for y in terms of x or for A in terms of t without the derivative.
To solve (1) we just integrate with respect to x.
Cx
yxdx
dy
2
2
However, we can’t integrate y w.r.t. x so (2) needs another method.
Solving Differential Equations
Before we see how to solve the equation, it’s useful to get some idea of the solution.
ydx
dye.g. (2)
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.
Solving Differential Equations
ydx
dy
We can now draw a curve through any point following the gradients.
Solving Differential Equations
ydx
dy
However, we haven’t got just one curve.
Solving Differential Equations
ydx
dy
The solution is a family of curves.
Can you guess what sort of equation these curves represent ?
ANS: They are exponential curves.
Solving Differential Equations
ydxdy Solvin
gWe use a method called “ Separating the Variables” and the title describes exactly what we do.
ydx
dydxdy
y
1
We rearrange so that x terms are on the right and y on the left.
Now insert integration signs . . .
dxdyy
1and integrate
Cxy ln
We can separate the 2 parts of the derivative because although it isn’t actually a fraction, it
behaves like one.
(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 don’t need a constant on both sides as they can be combined. I usually put it on the r.h.s.
Solving Differential Equations
ydx
dy
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 = . . . )
Cxy ln
A log is just an index, so
Cxyln Cxey
( We now have the exponential that we spotted from the gradient diagram. )
However, it can be simplified.
Solving Differential Equations
So, xkeyydx
dy
We can write as . Cxe Cx ee
where k is positive
This is usually written as where A is positive or negative.
xAey
Cxey
So, ydx
dy xAey 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.
Since is a constant it can be replaced by a single letter, k.
Ce
Solving Differential Equations
Changing the value of A gives the different curves we saw on the gradient diagram.
xAeyydx
dy
e.g. A = 2 gives
Solving Differential Equations
ydx
dyThe 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.
Solving Differential Equations
xydx
dycos2
e.g. 3 Solve the equation below giving the answer in the form .)(xfy
Solution:
Separating the variables:
dxxdyy
cos1
2
Insert integration signs:
dxxdyy cos2
Cxy
sin1
1
Integrate:
Cxy
sin1
Cxy
sin
1
Solving Differential Equations
xyxdx
dy
e.g. 4 Solve the equation below giving the answer in the form .)(xfy
Solution:
It’s no good dividing by y as this would give
xy
xdy
y
1 which is no help.
Instead, we take out x as a common factor on
the r.h.s., so )1( yxdx
dy
We can now separate the variables by dividing by
)1( y
xyxdx
dy
Solving Differential Equations
)1( yxdx
dy
Cx
y 2
1ln2
dxxdyy
1
1
Ay 1 2
2x
e
Cx
ey
2
2
1
Solving Differential Equations
)1( yxdx
dy
Cx
y 2
1ln2
dxxdyy
1
1
You may sometimes see this written as
2
exp2x
Ay 1 2
2x
e
Cx
ey
2
2
1
Solving Differential Equations
)1( yxdx
dy
Cx
y 2
1ln2
12
2
x
Aey
dxxdyy
1
1
Ay 1 2
2x
e
Cx
ey
2
2
1
Solving Differential Equations
dxxgdyyf )()(
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
dxxgdyyf )()(• The answer is often written
explicitly.• The solution is called the general solution.
Solving Differential Equations
Exercise
(a) xdx
dyx 2)1( 2 (b) 1
dx
dye y
2. Find the general solutions of the following equations leaving the answers in implicit form:(a) x
dx
dyy cossin b)
3. Find the equation of the curve given by the following equation and which passes through the given point.
0,0:2 2 yxexdx
dy y
1. Find the general solutions of the following equations giving your answers in the form :
)(xfy
( This is called a particular solution. )
yyxdx
dy 2
Solving Differential Equations
xdx
dyx 2)1( 2
Solutions:
dxx
xdy
21
2
Cxy )1ln( 2
(b) 1dx
dye y
dxdye y
Cxe y Cxy ln
dxxf
xf
)(
)(
Solving Differential Equations
2(a)
xdx
dyy cossin
dxxdyy cossin Cxy sincos
b) yyxdx
dy 2
dxxdyy
)1(1 2
Cxx
y 3
ln3
)1( 2 xydx
dy
Solving Differential Equations
0,0:2 2 yxexdx
dy y3.
dxxdye y
221
dxxdye y 22 C
xe y
3
2 3
0,0 yx C1
13
2 3
x
e yor 1
3
2ln
3
x
y
You might prefer to write as before you separate the variables.
ye ye
1
Solving Differential Equations
Solving Differential Equations
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.
Solving Differential Equations
ydxdy e.g. 1 Solving
We use a method called “ Separating the Variables” and the title describes exactly what we do.
ydx
dydxdy
y
1
We rearrange so that x terms are on the right and y on the left.
Now insert integration signs . . .
dxdyy
1and integrate
Cxy ln
(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.
Solving Differential Equations
ydx
dy
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 = . . . )
Cxy ln
A log is just an index, so
Cxyln Cxey
We usually use A.
So, xAeyydx
dy
We can write as . Since is a constant it can be replaced by a single letter.
Cxe Cx ee Ce
This can be simplified.
Solving Differential Equations
xydx
dycos
e.g. 2 Solve the equation below giving the answer in the form .)(xfy
Solution:
Separating the variables:
dxxdyy
cos1
Insert integration
signs:
dxxdyy cos1
Cxy sinlnIntegrate:Cxey sinxAey sin
Solving Differential Equations
xyxdx
dy
e.g. 3 Solve the equation below giving the answer in the form .)(xfy
Solution:
It’s no good dividing by y as this would give
xy
xdy
y
1 which is no help.
Instead, we take out x as a common factor on
the r.h.s. )1( yxdx
dy
We can now separate the variables by dividing by
)1( y
Solving Differential Equations
)1( yxdx
dy
Cx
y 2
)1ln(2
Cx
ey
2
2
1
2
2
1x
Aey
12
2
x
Aey
dxxdyy
1
1
Solving Differential Equations
dxxgdyyf )()(
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
dxxgdyyf )()(• The answer is often written
explicitly.• The solution is called the general solution.
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