Math

Differential equations Solutions

C.
Post + 2xy +"(9)
However, there is no function h(y) that would make this equal to N = x +2y, for were
this the case,
11.
+2xy + '(y) = x + xy,
which implies
W'(y) = -x -xy
But this is impossible since the right-hand side depends on x and is not a function of y
alone. We will see how to solve an equation such as this in Section 3.5.
.
EXERCISES 3.3
13. 2y sin(xy) dx + (2x sin(xy) + 3y) dy = 0,
y(0) = 1
z ?
dx = 0, y(O)=1
In Exercises 1-10, determine if the differential equation
is exact. If it is exact, find its solution.
1. (3x2 - 4y2) dx - (8xy - 12y3) dy = 0
2. (3xy + 4y2) dx + (5x2y + 2x2) dy = 0
3. (2xy + ye') dx + (x2 +et) dy = 0
4. (2xe* + x?yety - 2) dx +xety dy = 0
5. (2x cos y - x) dx + x2 sin y dy = 0
6. (y cos x + 3et cos y) dx +(sin x-3e" sin y) dy = 0
1-2xy
x2 - 2xy + 1
7. y'=
8. y' =
r2
x² - y²
ds el - 2t coss
dr -r cos e
10 9.
10.
di es - 12 sins
de
r + sin e
n Exercises 11-14, find the solution of the initial value
roblem.
1. 2xy dx + (x2 + 3y2) dy = 0, y(1) = 1
2xel - 3x2y
2. y'
x3 – x²e"
-, y(1)=0
14. 1
dyt
x² + y²
x2 + y2
15. Use Maple (or another appropriate software pack-
age) to graph the solution in Exercise 11.
16. Use Maple (or another appropriate software pack-
age) to graph the solution in Exercise 12.
17. Show a separable differential equation is exact.
18. Show the converse of Theorem 3.2 can be proved by
integrating with respect to y first.
y
19. Determine conditions on a, b, c, and d so that the
differential equation
ax + by
cx tdy
is exact and, for a differential equation satisfying
these conditions, solve the differential equation.
y exactress
+ N(x, y)
I equation
0.