1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lubasha [3.4K]
3 years ago
14

Show that ( 2xy4 + 1/ (x + y2) ) dx + ( 4x2 y3 + 2y/ (x + y2) ) dy = 0 is exact, and find the solution. Find c if y(1) = 2.

Mathematics
1 answer:
fredd [130]3 years ago
8 0

\dfrac{\partial\left(2xy^4+\frac1{x+y^2}\right)}{\partial y}=8xy^3-\dfrac{2y}{(x+y^2)^2}

\dfrac{\partial\left(4x^2y^3+\frac{2y}{x+y^2}\right)}{\partial x}=8xy^3-\dfrac{2y}{(x+y^2)^2}

so the ODE is indeed exact and there is a solution of the form F(x,y)=C. We have

\dfrac{\partial F}{\partial x}=2xy^4+\dfrac1{x+y^2}\implies F(x,y)=x^2y^4+\ln(x+y^2)+f(y)

\dfrac{\partial F}{\partial y}=4x^2y^3+\dfrac{2y}{x+y^2}=4x^2y^3+\dfrac{2y}{x+y^2}+f'(y)

f'(y)=0\implies f(y)=C

\implies F(x,y)=x^2y^3+\ln(x+y^2)=C

With y(1)=2, we have

8+\ln9=C

so

\boxed{x^2y^3+\ln(x+y^2)=8+\ln9}

You might be interested in
Need help with this math
olganol [36]

Hi,

Here is the solution.

3 years = 36 months

Leased amount for 3 years = $3000 + 36*350

= $3000 + $12600

Leased amount for 3 years = $15600

Residual value = $12000

Assume you didn't buy the car, you took it for lease, so your expense = $15600.

Now Assume you bought the car for $21,000.

After using 3 years, you can sell it for $12000.

Your expense = $21,000 - $12,000 = $9000.

Conclusion:

It is less expensive to buy the car.

The cost for buying the car and selling it after 3 years = $9000.

Hope you will understand the steps. Thank you :)

3 0
3 years ago
Do you think you know the answer comment it down
Nuetrik [128]

Answer:

for four right angles, should be only square.

for all sides congruent, square and rhombus

for 2 pairs of parallel sides, parallelogram and rhombus and square.

for only 1 pair of congruent sides, parallelogram and rhombus

Step-by-step explanation: doggo has asian smarts, not bsian

5 0
2 years ago
I need help with some of the history of math​
brilliants [131]
It’s the first option :)
5 0
3 years ago
Read 2 more answers
F(x)=x2 -x and g(x)=12-x2
jeka94

Answer are in pictures below

6 0
3 years ago
Which equation can be used to find the solution of (1/2)^2x = 32?
stira [4]

-2x=5 is the right answer

6 0
3 years ago
Other questions:
  • HELP ASAP!! Lesson 4: Perimeters and Areas of Similar Figures
    13·1 answer
  • If I make $200 a month and I needed to save $10,680 how long would it take
    13·2 answers
  • Write the expression below in standard form 3h-2(1+4h)
    12·1 answer
  • What is the solution of 3x + 5 = 2x – 7? the x-coordinates of the intersection point of the lines y = 3x + 5 and y = 2x – 7 the
    10·2 answers
  • What is the product?<br> 3 x [-4 5 -7]
    12·1 answer
  • The average weights of the dogs are the same for the morning and afternoon groups. Which group of dogs would be easiest to walk
    15·2 answers
  • What is the probability that a student's eye color is brown given that the student wears glasses?
    15·1 answer
  • The sides of a triangle are a + 3 ¼, 2a, and 7 ½ - a. If a = 4 1/8, what is the length of each of the three sides? Also, find th
    10·2 answers
  • What do people need to earn income?<br>​
    5·1 answer
  • Marcus rolled a number cube and recorded his results in the table. What is the experimental probability that the next roll is a
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!