Differential Equations

Restricting the Domain of the Solution

Sketch the solution of the differential equation on the given slope field:

xdx

dy 1

1)1( y

x

y

Note that while there are slopes to the left of the y-axis, our sketch can’t ever get that far because of the vertical slope at x = 0

Because the domain can only be one interval that contains the initial value.

Find the solution of the differential equation:

xdx

dy 1 1)1( y

x

dxdy Cxy ln

C )1ln(1 1C

1ln xy

…and the domain is…

0x

Plug in initial value (1, 1)

x

y

Because the domain can only be one interval that contains the initial value.

Sketch the solution of the differential equation on the given slope field:

xdx

dy 1

Note that while there are slopes to the left of the y-axis, our sketch can’t ever get that far because of the vertical slope at x = 0

1ln xy

0x

x

y

Sketch the solution of the differential equation on the given slope field:

2

3

dy

dx y

(1) 4y

Note that because we are dealing with positive slopes only, the sketch must stop just before the x-axis

One way to see why this is the domain would be to graph our solution

Find the solution of the differential equation:

2

3

dy

dx y (1) 4y

3

2y dy dx

3

2y x C

7C

…and the domain is…

7x

Plug in initial value (1, 4)

2

3y x C

2

34 1 C 8 1 C

2

37y x

x

y

One way to see why this is the domain would be to graph our solution

Sketch the solution of the differential equation on the given slope field:

Note that to the left of –7, the solution graph crosses the slopes instead of following them.

7x

2

37y x

And the derivative is

undefined for y = 0 so we must exclude x = –7

2

3

dy

dx y

x

y

Sketch the solution of the differential equation on the given slope field:

Note that to the left of –7, the solution graph crosses the slopes instead of following them.

7x

2

37y x

…and we would get negative slopes for x < –7 which contradicts the slope field

2

3

dy

dx y

Because derivative is

undefined for y = 0, we must exclude x = –7

x

y

The domain to any solution to a differential equation is the largest open interval containing the initial value.

It also needs to satisfy any restrictions on the derivative as well

When sketching the solution, the graph should never go against the slope field.

Like a continuous function, you should never have to pick your pencil up off the paper when sketching the solution