1 dy . . - = -y - sm xdx
....
3 2
0
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ '.' , ' . ' , โข I " , " " " " .... ...... ..... ..... , "
/ / 1 1 / / /
- 1 I ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ
I I I I I I \ I I I I I I I I I I I 1.1 I I . I I . 1 I I I I I I I I , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
1 1 1 1 1 1 1 1 / / / / "" "" "" ""' / /
- 2 ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ๏ฟฝ ๏ฟฝ ๏ฟฝ j ๏ฟฝ }.; ๏ฟฝ ;๏ฟฝ ๏ฟฝ ๏ฟฝI I I I I I I I I 1 1 1 / / / 1 / 1I I I I I I I I I I 1 1 1 1 1 1 1 1
-๏ฟฝ3 -2 - 1 0 2
dy 2. dx = x + y
- - / / / / 1 1 1- - - / / / / / /
2 ""-= ::: :::: ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ ๏ฟฝ
x
I I I I I I II I I I I I I lei I . I Ie I I I I I I I I I I I I I I I I I I I I I I I I I I / / 1 1 1 1 1 1 1/ / 1 1 1 1 1 1 1/ / / / / 1 1 1 1
3
.... o ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ-+--๏ฟฝ๏ฟฝ+ยฅ๏ฟฝ๏ฟฝ
- 1
- 2
3 dy . . - = y - sm xdx
2
I I I I I I I I I I I I I I I I I I Ie I lei I I I I I I I I I I I I I I I I I I I I
x
I I I I 1 I 1 1 11 1 / / / / 1 1 1I I ,. / / /., / I / / / / / / / / /
/ I I I I I I / I / 1 1 1 1 1 1 / // / / I / / / / .;/๏ฟฝ-๏ฟฝ๏ฟฝ
- 1 " ..... ...... ...... ..... , , '
-2 \ ๏ฟฝ ๏ฟฝ๏ฟฝ ๏ฟฝ ๏ฟฝ๏ฟฝ ๏ฟฝ ๏ฟฝ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ -๏ฟฝ3 -2 - 1 0
x 2
3
3
dy 4. - = x - ydx
2
- 1
-2 : :: :': :: ;๏ฟฝ ๏ฟฝ j - - - / / / / 1 1- - / / / / 1 1 1
-๏ฟฝ3 -2 - 1
dy 5. - = Y - x + 1dx
I I 2 I I I I
๏ฟฝ.๏ฟฝ I I .... 0
I I
o 2 x
- - - ..... , ' \ \ \ ./ - - ..... , ' \ \ \ \
- 1
- 2
- 1
dy 6. - = x - y + ldx
2
\ I I I I I \ \ \ I I I I 1 \ ' 1 .1 I . \ \ . ' \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ " \ \ \ \ \ \ " " .....
o x
- ..... " " \ '.' \ \ ...... , ' \ \ \ \ \ \ " \ \ \ \ \ \ \, \ \ \ \ \ \ \ \ , 1ยท 1 I I e.. I I , I I I I I \ \ I , \ I I I I \ I I
2
, ... , \ \ ' , ...... -\ \ \ \ ' , ..... - - ./
\ ,
- 1 : : _ _ ๏ฟฝ ; ๏ฟฝ ๏ฟฝ j- - - / / / 1 1 1
-2 :: ; ๏ฟฝ๏ฟฝ ๏ฟฝ ๏ฟฝ๏ฟฝ ๏ฟฝ ๏ฟฝ / / / / 1 1 1 1 /./ / I I I I I I I
-๏ฟฝ3 -2 - 1 0 x
3
3
Differential Equation - Spring 2015 - Classwork 1Instructor: Emil Sargsyan
Name: _____________________________
In problems 1-6 one of the solutions of the differentialequation is given. Sketch additional solutions through the given points.
7. Suppose the differential equation ๐๐๐๐๐๐๐๐
= ๐๐(๐ฅ๐ฅ,๐ฆ๐ฆ) has ๐๐(๐ฅ๐ฅ) as its solution. We know that the slope of the graph of ๐๐ at the point (๐ฅ๐ฅ,๐ฆ๐ฆ) is the product of ๐ฅ๐ฅ and ๐ฆ๐ฆ. What is ๐๐(๐ฅ๐ฅ,๐ฆ๐ฆ)?
8. (a) Sketch the slope field of the differential equation ๐๐๐๐๐๐๐๐
= ๐ฆ๐ฆ. The vectors donโt have to be precise, and be sure to produce the vectors at the positions (0,0), (1,0), (0,1), (-1,0), (0,-1),(0,3), (3,0), (0,-3), (-3,0).
(b) Solve the differential equation ๐ฆ๐ฆโฒ = ๐ฆ๐ฆ
(c) Sketch the curve ๐ฆ๐ฆ(๐ฅ๐ฅ) if the initial value is ๐ฆ๐ฆ(0) = 0.5.
(The curve does not have to be precise.)