DIFFERENTIAL EQUATIONS

In most cases, the family of functions will depend in
some way on a constant C, and the graphs of these
functions will form a family of curves that fill up the (t,
y) plane, but do not touch each other, as in the
following figure.

Example:

The general solution of

is

Graph of solution

has more than one solution

. One is simply y(x) = 0, the second one is obtained by

separating the variables. This leads to

The two solutions y = 0 and y = (x/5)

5

both satisfy the initial condition y(0) = 0

4

1

5

/

4

5

4

)'

(











y

y

y

f

The derivative of the previous example is not continuous

SEPARABLE EQUATIONS

1.

)

(

)

(

'

,

)

(

)

(

y

h

y

K

where

C

t

G

y

K







We find the antiderivatives of both sides

If it is possible, solve for y as a function of x y(x).

The general solution is the family of all solutions found above.
It will usually depend on C.

If an initial value y

0

(x) = y

0

is given, use it to find the constant C and

the particular solution of the problem.

1.cd

2.

3.

Example 1

Example 2

FIRST ORDER LINEAR

DIFFERENTIAL EQAUTIONS

(*)

(*)

(*)

The above equation has separable variables, because it
can be written

0

)

(

)

(

'









y

x

p

dx

dy

y

x

p

y

y

x

p

dx

dy

)

(





VARIATION OF CONSTANTS

c

t

t

C

t

t

C

t

t

t

C

RHS

LHS

t

RHS

t

t

C

t

t

C

t

t

t

C

t

t

C

y

t

y

LHS















































6

)

(

)

(

'

)

(

'

)

(

'

)

(

3

)

3

)(

(

)

(

'

3

'

6

5

2

3

2

3

3

1

4

3

1

The particular solution is x(t) = C t

-3

,

Step 2

Take

3

)

(

)

(





t

t

C

t

y

3

3

3

6

1

6

1

)

6

(

)

(

t

c

t

t

c

t

t

y









General solution is