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
Gelneren [198K]
2 years ago
8

Solve the differential equation (2x+y)dx+(x−2y)dy = 0

Mathematics
1 answer:
ruslelena [56]2 years ago
5 0

Answer:

y(x) = 1/2 (x - sqrt(5 x^2 + c_1)) or y(x) = 1/2 (x + sqrt(5 x^2 + c_1))

Step-by-step explanation:

Solve 2 x + y(x) + (dy(x))/(dx) (x - 2 y(x)) = 0:

Let P(x, y) = 2 x + y and Q(x, y) = x - 2 y.

This is an exact equation, because (dP(x, y))/(dy) = 1 = (dQ(x, y))/(dx).

Define f(x, y) such that (df(x, y))/(dx) = P(x, y) and (df(x, y))/(dy) = Q(x, y).

Then, the solution will be given by f(x, y) = c_1, where c_1 is an arbitrary constant.

Integrate (df(x, y))/(dx) with respect to x in order to find f(x, y):

f(x, y) = integral(2 x + y) dx = x^2 + x y + g(y) where g(y) is an arbitrary function of y.

Differentiate f(x, y) with respect to y in order to find g(y):

(df(x, y))/(dy) = d/(dy) (x^2 + y x + g(y)) = x + (dg(y))/(dy)

Substitute into (df(x, y))/(dy) = Q(x, y):

x + (dg(y))/(dy) = x - 2 y

Solve for (dg(y))/(dy):

(dg(y))/(dy) = -2 y

Integrate (dg(y))/(dy) with respect to y:

g(y) = integral-2 y dy = -y^2

Substitute g(y) into f(x, y):

f(x, y) = x^2 - y^2 + y x

The solution is f(x, y) = c_1:

x^2 - y^2 + y x = c_1

Solve for y:

y(x) = 1/2 (x - sqrt(5 x^2 - 4 c_1)) or y(x) = 1/2 (x + sqrt(5 x^2 - 4 c_1))

Simplify the arbitrary constants:

Answer:  y(x) = 1/2 (x - sqrt(5 x^2 + c_1)) or y(x) = 1/2 (x + sqrt(5 x^2 + c_1))

You might be interested in
Maci and I are making a small kite. Two sides are 10". Two sides are 5". The shorter diagonal is 6". Round all your answers to t
Art [367]

Answer:

A. 4".

B. Approximately 9.54".

C. Approximately 13.54".

Step-by-step explanation:

Please find the attachment.

Let x be the distance from the peak of the kite to the intersection of the diagonals and y be the distance from the peak of the kite to the intersection of the diagonals.

We have been given that two sides of a kite are 10 inches and two sides are 5 inches. The shorter diagonal is 6 inches.

A. Since we know that the diagonals of a kite are perpendicular and one diagonal (the main diagonal) is the perpendicular bisector of the shorter diagonal.

We can see from our attachment that point O is the intersection of both diagonals. In triangle AOD the side length AD will be hypotenuse and side length DO will be one leg.

We can find the value of x using Pythagorean theorem as:

(AO)^2=(AD)^2-(DO)^2

x^{2}=5^2-3^2

x^{2}=25-9

x^{2}=16

Upon taking square root of both sides of our equation we will get,

x=\sqrt{16}

x=\pm 4

Since distance can not be negative, therefore, the distance from the peak of the kite to the intersection of the diagonals is 4 inches.

B. We can see from our attachment that point O is the intersection of both diagonals. In triangle DOC the side length DC will be hypotenuse and side length DO will be one leg.

We can find the value of y using Pythagorean theorem as:

(OC)^2=(DC)^2-(DO)^2

Upon substituting our given values we will get,

y^2=10^2-3^2

y^2=100-9

y^2=91

Upon taking square root of both sides of our equation we will get,

y=\sqrt{91}

y\pm 9.539392

y\pm\approx 9.54

Since distance can not be negative, therefore, the distance from intersection of the diagonals to the top of the tail is approximately 9.54 inches.

C. We can see from our diagram that the length of longer diagram will be the sum of x and y.

\text{The length of the longer diagonal}=x+y

\text{The length of the longer diagonal}=4+9.54

\text{The length of the longer diagonal}=13.54

Therefore, the length of longer diagonal is approximately 13.54 inches.

3 0
3 years ago
The diameter of the average cell in the human body is 1 x 10-5 meters. The diameter of a grain of sand is about 2 x 10-4 meters.
Ad libitum [116K]
1x-10-4 divided by 25
6 0
3 years ago
you put $3000 in a CD (certificate of deposit) at the promotional rate of 5.6% simple interest. How long will it take to earn $3
Wewaii [24]

Answer:

1 year 9 months

Step-by-step explanation:


6 0
3 years ago
What is the solution for -10 < x - 9?
evablogger [386]

- 10 < x - 9
ANSWER
x >  - 1
7 0
3 years ago
Read 2 more answers
What is an equation of the like with a slope of 1/4 and a y intercept of -2
SVEN [57.7K]
Y = mx + c is the general equation of a line. m is the slope and c is the y intercept, by substituting these values in you get y = 1/4 x -2
3 0
3 years ago
Read 2 more answers
Other questions:
  • Describe the trend in the scatter plot.
    15·1 answer
  • Jessica traveled to Portland by car. Going there took 3 hours, and the return trip lasted 6 hours. Jessica averaged a speed of 4
    13·1 answer
  • What is 5 expressed as a percent? out of 100
    10·2 answers
  • Is 17 feet bigger than 804 inches
    6·2 answers
  • Midge walked 7/8 mile Monday and 4/5 mile on Tuesday. let x = how much farther she walked on Monday.
    12·2 answers
  • The chain of the Hawaiian Islands can be used to calculate the rate of movement of the Pacific Plate over the last few million y
    13·1 answer
  • Help ASAP plzzzz!!!! T^T
    10·1 answer
  • Find the Value of X in simplest radical form and Find the measure of angle A.​
    12·1 answer
  • Which table and graph represent the equation y = 5x? A. Table B and graph B B. Table A and graph A C. Table A and graph B D. Tab
    7·2 answers
  • I need help asap please? and this is for the math question? what is 77x+93=360? put in a fraction and decimal and you can round
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!