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
ankoles [38]
3 years ago
9

Consider the implicit differential equation

20%2898%20xy%5E%7B2%7D%20%2B50%20x%5E%7B2%7D%20%29dy%3D0" id="TexFormula1" title="(49 y^{3} + 45 xy) dx + (98 xy^{2} +50 x^{2} )dy=0" alt="(49 y^{3} + 45 xy) dx + (98 xy^{2} +50 x^{2} )dy=0" align="absmiddle" class="latex-formula">. For the integrating factor x^{p}  y^{q} of this equation, find p and q. Now multiply the equation by the integrating factor x^{p}  y^{q} that you have found and then integrate the resulting equation to get a solution in implicit form.
Mathematics
1 answer:
BaLLatris [955]3 years ago
8 0
We're looking for an integrating factor \mu(x,y)=x^py^q such that

\mu\underbrace{(49y^3+45xy)}_M\,\mathrm dx+\mu\underbrace{(98xy^2+50x^2)}_N\,\mathrm dy=0

is exact, which would require that

(\mu M)_y=(\mu N)_x
(49x^py^{q+3}+45x^{p+1}y^{q+1})_y=(98x^{p+1}y^{q+2}+50x^{p+2}y^q)_x
49(q+3)x^py^{q+2}+45(q+1)x^{p+1}y^q=98(p+1)x^py^{q+2}+50(p+2)x^{p+1}y^q
\implies\begin{cases}49(q+3)=98(p+1)\\45(q+1)=50(p+2)\end{cases}\implies p=\dfrac52,q=4

You can verify that (\mu M)_y=(\mu N)_x if you'd like. With the ODE now exact, we have a solution F(x,y)=C such that

F_x=\mu M
F=\displaystyle\int(49y^3+45xy)x^{5/2}y^4\,\mathrm dx
F=10x^{9/2}y^5+14x^{7/2}y^7+f(y)

F_y=\mu N
50x^{9/2}y^4+98x^{7/2}y^6+f'(y)=98x^{7/2}y^2+50x^{9/2}y^4
f'(y)=0
\implies f(y)=C

and so the general solution is

F(x,y)=10x^{9/2}y^5+14x^{7/2}y^7=C
You might be interested in
You earn $50 per day for babysitting plus $0.75 per mile that she has to drive for an equation to represent the situation and de
tatuchka [14]
50 + (.75 x d)

D = distance traveled by in miles
8 0
3 years ago
Why do you think the Court ruled differently in New York Times Co. v. United States (1971) than it did in Schenck v. United Stat
Marina CMI [18]

Answer:

The case reached the Supreme Court in June 1971. The Supreme Court has, at times, ruled that the government can restrict speech that presents a “clear and present danger.”

5 0
3 years ago
Which equation matches the graph?
DIA [1.3K]

Answer:

y=3x-2

Step-by-step explanation:

8 0
2 years ago
Read 2 more answers
In a class of 34 students, 19 of them are girls.
cluponka [151]

Answer:

55.8

Step-by-step explanation:

8 0
3 years ago
Please help me <br> I’ll mark brainiest
Lubov Fominskaja [6]

Answer:

which number

every of them?

8 0
2 years ago
Other questions:
  • PLEASE HELP!!!!!
    12·2 answers
  • Use a distributive property to write an expression equivalent to 5 x(3+4)
    7·1 answer
  • What's the closest integer to the square root of 99?
    5·1 answer
  • Consider the sequence of steps to solve the equation: 3(x − 4) + 5x = 9x − 36 Given ⇒ 3(x − 4) + 5x = 9x − 36 Step 1 ⇒ 3x − 12 +
    11·1 answer
  • For the students in your school, the mean score on a state-wide standardized test was 116. Is the mean score a parameter or a st
    9·1 answer
  • Please help and explain ( Brainliest for answer!)
    14·1 answer
  • I need the answer fast
    9·2 answers
  • Find the height of a parallelogram with a base 24 inches and the area of 432 square inches
    6·1 answer
  • Subtract. Write your answer as a mixed number in simplest form.<br><br> 8 1/8 - 3 3/8
    9·2 answers
  • Circumference of the circle<br> R=9in
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!