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Differential Equations Homework: Exact Differential Equations, Exams of Algebra

Rice UniversityAlgebra

Georgia Institute of Technology, Fall 2015. Homework 2. Exact Differential Equations. For each ODE below, determine whether or not it is exact.

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Math 2552 - Differential Equations
Sections F1 – F4; L1 – L4
Georgia Institute of Technology, Fall 2015
Homework 2
Exact Differential Equations
For each ODE below, determine whether or not it is exact. If it is exact, solve it. (No need to give an
interval of validity.)
1. (5x+ 4y)dx + (4x−8y3)dy = 0.
2. sin y−ysin xdx +cos x+xcos y−ydy = 0.
3. cos(xy)−xy sin(xy)dx −x2sin(xy)dy = 0.
4. yexy dx + (2y−xexy )dy = 0.
5. 1 + ln(xy)dx +x
ydy = 0.
6. (2y2x−3) dx + (2yx2+ 4) dy = 0.
7. 2y−
1
x+cos(3x)dy
dx +y
x2−4x3+3ysin(3x)=0.
8. (x3+y3)dx + 3xy2dy = 0.
9. y3
−y2sin x−xdx +3xy2+ 2ycos xdy = 0.
10. yln y−e−xydx +1
y+xln ydy = 0.
11. 2x
ydx −
x2
y2dy = 0.
12. 1
x−
y
x2+y2dx +x
x2+y2dy = 0.
.......................................................................................................
Solve each of the initial value problems below:
13. (x+y)2dx + (2xy +x2
−1) dy = 0; y(1) = 1.
14. (ex+y)dx + (2 + x+yey)dy = 0; y(0) = 1.
15. (y2cos x−3x2y−2x)dx + (2ysin x−x3+ ln y)dy = 0; y(0) = e.
16. 1
1 + y2+ cos x−2xydy
dx =y(y+ sin x); y(0) = 1.
.......................................................................................................
For each of the ODEs below, find the value of ksuch that the equation is exact, and then solve the equation.
17. (y3+kxy4
−2x)dx + (3xy2+ 20x2y3)dy = 0.
18. (2x−ysin(xy) + ky4)dx −(20xy3+xsin(xy)) dy = 0.
19. (2xy2+yex)dx + (2x2y+kex
−1) dy = 0.
20. (6xy3+ cos y)dx + (k x2y2
−xsin y)dy = 0.
.......................................................................................................
Find a function M(x, y) such that the ODE is exact:
21. M(x, y)dx +xexy + 2xy +1
xdy = 0.
Find a function N(x, y) such that the ODE is exact:
22. √y
√x+x
x2+y!dx +N(x, y)dy = 0.
.......................................................................................................
For the ODEs below, verify that the given µ(x, y) is an integrating factor, and use it to solve the equation:
23. y(x+y+ 1) dx + (x+ 2y)dy = 0; µ(x, y) = ex.
24. (−xy sin x+ 2ycos x)dx + 2xcos x dy = 0; µ(x, y) = xy.

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Math 2552 - Differential Equations Sections F1 – F4; L1 – L Georgia Institute of Technology, Fall 2015

Homework 2

Exact Differential Equations

For each ODE below, determine whether or not it is exact. If it is exact, solve it. (No need to give an interval of validity.)

  1. (5x + 4y) dx + (4x − 8 y^3 ) dy = 0.

sin y − y sin x

dx +

cos x + x cos y − y

dy = 0.

[

cos(xy) − xy sin(xy)

]

dx − x^2 sin(xy) dy = 0.

  1. yexy^ dx + (2y − xexy^ ) dy = 0.

1 + ln(xy)

dx + x y dy = 0.

  1. (2y^2 x − 3) dx + (2yx^2 + 4) dy = 0.

2 y −

x +cos(3x)

) (^) dy dx

y x^2 − 4 x^3 +3y sin(3x) = 0.

  1. (x^3 + y^3 ) dx + 3xy^2 dy = 0.

y^3 − y^2 sin x − x

dx +

3 xy^2 + 2y cos x

dy = 0.

y ln y − e−xy^

dx +

y

  • x ln y

dy = 0.

2 x y dx − x^2 y^2 dy = 0.

x

y x^2 + y^2

dx + x x^2 + y^2 dy = 0.

....................................................................................................... Solve each of the initial value problems below:

  1. (x + y)^2 dx + (2xy + x^2 − 1) dy = 0; y(1) = 1.
  2. (ex^ + y) dx + (2 + x + yey^ ) dy = 0; y(0) = 1.
  3. (y^2 cos x − 3 x^2 y − 2 x) dx + (2y sin x − x^3 + ln y) dy = 0; y(0) = e.

1 + y^2

  • cos x − 2 xy

) (^) dy dx = y(y + sin x); y(0) = 1.

....................................................................................................... For each of the ODEs below, find the value of k such that the equation is exact, and then solve the equation.

  1. (y^3 + kxy^4 − 2 x) dx + (3xy^2 + 20x^2 y^3 ) dy = 0.
  2. (2x − y sin(xy) + ky^4 ) dx − (20xy^3 + x sin(xy)) dy = 0.
  3. (2xy^2 + yex) dx + (2x^2 y + kex^ − 1) dy = 0.
  4. (6xy^3 + cos y) dx + (kx^2 y^2 − x sin y) dy = 0. .......................................................................................................

Find a function M (x, y) such that the ODE is exact:

  1. M (x, y) dx +

xexy^ + 2xy +

x

dy = 0.

Find a function N (x, y) such that the ODE is exact:

y √ x

  • x x^2 + y

dx + N (x, y) dy = 0.

For the ODEs below, verify that the given μ(x, y) is an integrating factor, and use it to solve the equation:

  1. y(x + y + 1) dx + (x + 2y) dy = 0; μ(x, y) = ex.
  2. (−xy sin x + 2y cos x) dx + 2x cos x dy = 0; μ(x, y) = xy.