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
yawa3891 [41]
2 years ago
13

Find general solutions of the differential equation. Primes denote derivatives with respect to x.

Mathematics
1 answer:
zepelin [54]2 years ago
5 0

Answer:

\mathbf{3x^2y^3+2xy^4=C}

Step-by-step explanation:

From the differential equation given:

6xy^3 +2y ^4 +(9x^2y^2+8xy^3) y' = 0

The equation above can be re-written as:

6xy^3 +2y^4 +(9x^2y^2+8xy^3)\dfrac{dy}{dx}=0

(6xy^3 +2y^4)dx +(9x^2y^2+8xy^3)dy=0

Let assume that if function M(x,y) and N(x,y) are continuous and have continuous first-order partial derivatives.

Then;

M(x,y) dx + N (x,y)dy = 0; this is exact in R if and only if:

\dfrac{{\partial M }}{{\partial y }}= \dfrac{\partial N}{\partial  x}}} \ \ \text{at each point of R}

relating with equation M(x,y)dx + N(x,y) dy = 0

Then;

M(x,y) = 6xy^3 +2y^4\  and \ N(x,y) = 9x^2 y^2 +8xy^3

So;

\dfrac{\partial M}{\partial y }= 18xy^3 +8y^3

       \dfrac{\partial N}{\partial y }

Let's Integrate \dfrac{\partial F}{\partial x}= M(x,y) with respect to x

Then;

F(x,y) = \int (6xy^3 +2y^4) \ dx

F(x,y) = 3x^2 y^3 +2xy^4 +g(y)

Now, we will have to differentiate the above equation with respect to y and set \dfrac{\partial F}{\partial x}= N(x,y); we have:

\dfrac{\partial F}{\partial y} = \dfrac{\partial}{\partial y } (3x^2y^3+2xy^4+g(y)) \\ \\ = 9x^2y^2 +8xy^3 +g'(y) \\ \\ 9x^2y^2 +8xy^3 +g'(y) =9x^2y^2 +8xy^3 \\ \\ g'(y) = 0  \\ \\ g(y) = C_1

Hence, F(x,y) = 3x^2y^3 +2xy^4 +g(y)  \\ \\ F(x,y) = 3x^2y^3 + 2xy^4 +C_1

Finally; the general solution to the equation is:

\mathbf{3x^2y^3+2xy^4=C}

You might be interested in
The scores for a basketball team are 72, 84, 102, 73, 90 and 83. the scores for the next two matches bring the average up to 90.
aniked [119]
72 + 84 + 102 + 73 + 90 + x
OVER
9
EQUALS 90
6 0
3 years ago
Read 2 more answers
The amount​ (in megawatts) of energy generated in a country from wind power can be modeled by the function ​W(t)equalsw0​(1.90​)
Ad libitum [116K]

Answer with Step-by-step explanation:

We are given that

W(t)=W_0(1.90)^t

Where t corresponds to the year 2000=0

a.W(0)=2540MW

Substitute the values

2540\times 10^6=W_0(1.90)^0=W_0

W_0=2540MW

b.In  2004

t=4

W(4)=2540(1.90)^4

W(4)=33101.534 MW

c.In 2013

t=13

W(13)=2540(1.9)^{13}=10.68\times 10^6MW

6 0
3 years ago
The product of two positive numbers is greater than the sum of the two numbers
andreyandreev [35.5K]

Answer:

no cause it is negative.

two positives make a negative

7 0
3 years ago
Is it a function? {(3,2),(0,1),(-2,1),(3,-2)}
11111nata11111 [884]

Answer:

no

Step-by-step explanation:

to know if it is a function or not look at the x coordinate if any of the x coordinates repeat then its not a function but of any of the x coordinates don't repeat than it is a function

5 0
3 years ago
Read 2 more answers
Which equation has infinitely many solutions?
ipn [44]

Answer:

Equation C. 5.1 + 2y + 1.2 = -2 + 2y + 8.3

Step-by-step explanation:

Equation C is the only equation in the list in which the terms that contain the unknown "y" on each side of the equal sign are identical, therefore when solving for this unknown and trying to group them on one side, they go away, leaving us with a relationship among numerical values that is always true:

5.1 + 2y + 1.2 = -2 + 2y + 8.3

5.1 + 1.2 = -2 + 8.3

6.3 = 6.3

Then this equation is true for any value of the unknown y, and y- can adopt infinite number of values, independent of which the equation will always be a true statement (giving thus infinite number of solutions).

4 0
3 years ago
Other questions:
  • 14x + 6 = 5x - 3 with solution
    13·1 answer
  • What is the arc measure of DBC in degree
    14·1 answer
  • The dewey civic center has 2500 seats. Tickets for 72% of the yotal number of seats were sold. How many tickets were sold?
    8·1 answer
  • La razon es la.división de dos números
    15·1 answer
  • What is the solution to the equation 9 –3x ≈ 7 ?
    7·1 answer
  • Help me please quickly.
    7·1 answer
  • Due now please help!
    14·2 answers
  • What is the range of possible sizes for x?
    11·1 answer
  • The complement of an angle is 32 more than three times the angle. what are the angles
    13·2 answers
  • The ordered pairs below represent a relation between x and y.
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!