Differential Equations

Graphing Solutions

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Solving a first-order D.E. explicitly (finding a formula solution) can be difficult or impossible, depending on the equation. For example, take a look at the following example:

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

This D.E. is first-order, but it is nonlinear and not separable. Basically we can’t solve it explicitly by any elementary methods.

Solving a first-order D.E. explicitly (finding a formula solution) can be difficult or impossible, depending on the equation. For example, take a look at the following example:

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

This D.E. is first-order, but it is nonlinear and not separable. Basically we can’t solve it explicitly by any elementary methods.

We can, however, get a graphical solution by using the D.E. directly, instead of looking for a formula solution. The graph we obtain is called a SLOPE FIELD.

How to draw a Slope Field

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

The most direct method uses the D.E. to find the slope of the solution one point at a time, then draw a short line segment with that slope at the chosen point.

How to draw a Slope Field

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

The most direct method uses the D.E. to find the slope of the solution one point at a time, then draw a short line segment with that slope at the chosen point.

Let’s do this for the given equation.Make a table of values.

t y y’

-2 -2

-2 -1

-2 0

-2 1

-2 2

-1 -2

-1 -1

-1 0

-1 1

-1 2

0 -2

0 -1

0 0

0 1

0 2

1 -2

1 -1

1 0

1 1

1 2

2 -2

2 -1

2 0

2 1

2 2

How to draw a Slope Field

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

The most direct method uses the D.E. to find the slope of the solution one point at a time, then draw a short line segment with that slope at the chosen point.

Let’s do this for the given equation.Make a table of values.

The calculations should look something like this:

9.3e)2()2,2('y

6.3e)2()1,2('y

3e)2()0,2('y

3.1e)2()1,2('y

4.3e)2()2,2('y

)2(2

)1(2

)0(2

)1(2

)2(2

t y y’

-2 -2 -3.4

-2 -1 1.3

-2 0 3

-2 1 3.6

-2 2 3.9

-1 -2

-1 -1

-1 0

-1 1

-1 2

0 -2

0 -1

0 0

0 1

0 2

1 -2

1 -1

1 0

1 1

1 2

2 -2

2 -1

2 0

2 1

2 2

How to draw a Slope Field

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Here is our table all filled in.This should give us a rough picture of the solutions.

t y y’

-2 -2 -3.4

-2 -1 1.3

-2 0 3

-2 1 3.6

-2 2 3.9

-1 -2 -6.4

-1 -1 -1.7

-1 0 0

-1 1 0.6

-1 2 0.9

0 -2 -7.4

0 -1 -2.7

0 0 -1

0 1 -0.4

0 2 -0.1

1 -2 -6.4

1 -1 -1.7

1 0 0

1 1 0.6

1 2 0.9

2 -2 -3.4

2 -1 1.3

2 0 3

2 1 3.6

2 2 3.9

How to draw a Slope Field

y2 ety

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

A few possible solutions are sketched on the slope field.

t y y’

-2 -2 -3.4

-2 -1 1.3

-2 0 3

-2 1 3.6

-2 2 3.9

-1 -2 -6.4

-1 -1 -1.7

-1 0 0

-1 1 0.6

-1 2 0.9

0 -2 -7.4

0 -1 -2.7

0 0 -1

0 1 -0.4

0 2 -0.1

1 -2 -6.4

1 -1 -1.7

1 0 0

1 1 0.6

1 2 0.9

2 -2 -3.4

2 -1 1.3

2 0 3

2 1 3.6

2 2 3.9

How to draw a Slope Field - Isoclines

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Another method for drawing slope fields that can be more efficient is to use ISOCLINES. An isocline is a curve where the solution to the D.E. has constant slope. We find the isoclines by choosing a value for y’, then graphing the corresponding curve. Let’s try this method with the following example:

tty

'y

This example is a first-order, linear D.E., but it is not separable. It can be solved explicitly, and we will learn how to do that soon. We can get a graph of the solutions by drawing the slope field.

How to draw a Slope Field - Isoclines

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

If the D.E. is easily solved for the dependent variable then do so.

t)1'y(yt

ty'y

How to draw a Slope Field - Isoclines

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

If the D.E. is easily solved for the dependent variable then do so.

t)1'y(yt

ty'y

It looks like our isoclines will be straight lines through the origin with slope equal to (y’-1).

Choose a slope value for y’, draw the corresponding isocline, then put hash marks on it at the slope that you chose.

How to draw a Slope Field - Isoclines

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

If the D.E. is easily solved for the dependent variable then do so.

t)1'y(yt

ty'y

It looks like our isoclines will be straight lines through the origin with slope equal to (y’-1).

Choose a slope value for y’, draw the corresponding isocline, then put hash marks on it at the slope that you chose.

ty2'y

0y1'y

ty0'y

t2y1'y

t3y2'y

How to draw a Slope Field - Isoclines

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

If the D.E. is easily solved for the dependent variable then do so.

t)1'y(yt

ty'y

The picture should look

something like this when you are done:

Autonomous Differential Equations

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

If the independent variable does not appear in a D.E. it is called AUTONOMOUS. A few examples:

ye

'y

)ytan('y

2yy'y

yy'y

y'y

y

2

2

Autonomous Differential Equations

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

If the independent variable does not appear in a D.E. it is called AUTONOMOUS. A few examples:

ye

'y

)ytan('y

2yy'y

yy'y

y'y

y

2

2

Notice that all of these equations are separable.Also, There could be an equilibrium solution where y’=0. This is the first thing you should do when you see an autonomous D.E. – find any equilibrium solutions.