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2 years ago

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))

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Question #2

RideAnS [48]

Answer:

First, the student performed what was inside of the parentheses and was exponential, which is correct. We know this because of the rules called order of operations, outlined by PEMDAS, which tells us to perform mathematical operations in the following order: parentheses, exponents, multiplication, division, addition, subtraction.

So by the rules I just outlined, next the student should have performed the multiplication, but instead performed subtraction, which is incorrect.

The following is the correct simplification of the expression:

23 - 8 times 2 + 3 by the power of 2

So first we should perform parentheses, but there are none. So we move to the next option in the list, which is exponents, which we do have (3^2).

23 - 8 * 2 + 3^2

23 - 8 * 2 + 9

Next in PEMDAS comes multiplication and division. This means the next step in simplifying the expression is to evaluate 8*2.

23 - 16 + 9

Next in PEMDAS is addition, so we should perform the operation -16 + 9.

23 - 7

Finally, the last step in PEMDAS is to simplify the remaining operation (subtraction).

Therefore, the student did the first step correctly but performed the second step incorrectly. The final answer should be 16 after simplification.

Hope this helps!

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2 years ago

Find the midpoint of the segment with the following endpoints.<br> (-2,7) and (-10, -1)

Lena [83]

The midpoint would be (-6,3)

3 0

3 years ago

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Find the distance between the points given.<br> (3, 4) and (6,8)

Damm [24]

Answer:

Slope of 3/4 (3,4) between points

Step-by-step explanation:

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3 years ago

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Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth.

marishachu [46]

Answer:

Thus, the two root of the given quadratic equation $x^2-6=-x$ is 2 and -3 .

Step-by-step explanation:

Consider, the given Quadratic equation, $x^2-6=-x$

This can be written as , $x^2+x-6=0$

We have to solve using quadratic formula,

For a given quadratic equation $ax^2+bx+c=0$ we can find roots using,

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ ...........(1)

Where, $\sqrt{b^2-4ac}$ is the discriminant.

Here, a = 1 , b = 1 , c = -6

Substitute in (1) , we get,

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$\Rightarrow x=\frac{-(1)\pm\sqrt{(1)^2-4\cdot 1 \cdot (-6)}}{2 \cdot 1}$

$\Rightarrow x=\frac{-1\pm\sqrt{25}}{2}$

$\Rightarrow x=\frac{-1\pm 5}{2}$

$\Rightarrow x_1=\frac{-1+5}{2}$ and $\Rightarrow x_2=\frac{-1-5}{2}$

$\Rightarrow x_1=\frac{4}{2}$ and $\Rightarrow x_2=\frac{-6}{2}$

$\Rightarrow x_1=2$ and $\Rightarrow x_2=-3$

Thus, the two root of the given quadratic equation $x^2-6=-x$ is 2 and -3 .

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3 years ago

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A recipe calls for 120 mL of milk for each 200 g of flour.Juan is going to use 1 pound of flour. Estimate the number cups of mil

Ratling [72]

Answer:

<u>1 cup</u> of milk Juan should use.

Step-by-step explanation:

Given:

A recipe calls for 120 ml of milk for each 200 g of flour.

Juan is going to use 1 pound of flour.

Now, to estimate the number of cups of milk.

Let the quantity of milk be $x.$

Now, to convert the pound of flour to gram.

1 pound = 453.59 g.

So, Juan is going to use 453.59 g of flour.

As, given recipe calls for 120 ml of milk for each 200 g of flour.

So, 120 ml is equivalent to 200 g.

Thus, $x$ is equivalent to 453.59 g.

Now, to solve by using cross multiplication method:

$\frac{120}{200} =\frac{x}{453.59}$

By cross multiplying we get:

$54430.8=200x$

Dividing both sides by 200 we get:

$272.154=x$

$x=272.154$

The quantity of milk = 272.154 ml.

Now, converting the quantity into cups by conversion factor:

1 ml = 0.004 cup.

So, 272.154 ml of milk is:

272.154 × 0.004

=1.088 cups.

Estimating the number = 1 cup.

Therefore, 1 cup of milk Juan should use.

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3 years ago