Worksheet 2: Sections 2.1 – 2.3
Considerations when sketching a slopefield
1) What happens when x = 0 and y = 0, i.e. along
the axes?
2) Suppose the right hand side of the equation only
contains the independent variable x. What can
you say about the slopefield?
3) Suppose the right hand side of the equation only
contains the dependent variable y. What can you
say about the slopefield?
4) What happens when x →  and when y → ?
dy
 0 , and how can you read that off
dx
the slopefield?
dy
6) What about when
is undefined?
dx
dy
7) Look at each quadrant to see if
is positive
dx
or negative.
5) When is
dy
 f ( y ) ),
dx
then draw the phase line, find critical points, and label each point as stable, unstable, or semistable.
Match each slopefield with the differential equations below. If the slopefield is autonomous (
_____1)
dy
 ( y  1)( y  3)
dx
_____2)
A)
6. Sketch the slopefield:
B)
dy
 y3  y2
dx
dy
 x y
dx
_____3)
C)
dy
y
dx
_____4)
D)
dy
x

dx x  2
Separable 1st–order ODE’s can be written in the form
dy
To solve, rewrite the equation in the form
 f  x, y   g  x   h y 
dx
dy
 g ( x)dx and integrate each side of the equation.
h( y )
Solve for y, if possible.
7)
dy
 y 1 / 2 (1  x ) , Initial Condition: y(0) = 1
dx
Linear 1st–order ODE’s may be written in the form
dy
 P( x)  y  Q( x)
dx
P ( x ) dx
To solve, multiply each term of the equation by an integrating factor of the form I(x) = e 
8) Write the equation in the form shown above, introduce the integrating factor, and solve: t ln t
9) Solve the linear 1st–order BVP. t
dr
 r  tet
dt
dy
 2 y  t 3 , Boundary Value: y(1) = 0
dt
For each of the following problems, first decide if the equation is linear, separable, or both. Try to find the
general solution.
10) y   e x  2 y , y
11) y   y  x 3
12) y  
13)
1 y2
cos 2 x
dy
 2 xy  ln( x)e x
dx
2