PLEASE HELP DUE IN 5 MINUTS

Find the general solution to 1/x dy/dx - 2y/x^2 = x cos x, y(pi) = pi^2

Finger [1]

Answer:

$\frac{y}{x^2}=\sin x+\pi$

Step-by-step explanation:

Consider linear differential equation $\frac{\mathrm{d} y}{\mathrm{d} x}+yp(x)=q(x)$

It's solution is of form $y\,I.F=\int I.F\,q(x)\,dx$ where I.F is integrating factor given by $I.F=e^{\int p(x)\,dx}$ .

Given: $\frac{1}{x}\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{2y}{x^2}=x\cos x$

We can write this equation as $\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{2y}{x}=x^2\cos x$

On comparing this equation with $\frac{\mathrm{d} y}{\mathrm{d} x}+yp(x)=q(x)$ , we get $p(x)=\frac{-2}{x}\,\,,\,\,q(x)=x^2\cos x$

I.F = $e^{\int p(x)\,dx}=e^{\int \frac{-2}{x}\,dx}=e^{-2\ln x}=e^{\ln x^{-2}}=\frac{1}{x^2}$ { formula used: $\ln a^b=b\ln a$ }

we get solution as follows:

$\frac{y}{x^2}=\int \frac{1}{x^2}x^2\cos x\,dx\\\frac{y}{x^2}=\int \cos x\,dx\\\\\frac{y}{x^2}=\sin x+C$

{ formula used: $\int \cos x\,dx=\sin x$ }

Applying condition: $y(\pi)=\pi^2$

$\frac{y}{x^2}=\sin x+C\\\frac{\pi^2}{\pi}=\sin\pi+C\\\pi=C$

So, we get solution as :

$\frac{y}{x^2}=\sin x+\pi$

4 0

4 years ago

Point G bisects the line segment FH. The length of FG is 16 less than 3 times the length of GH. What is the length of FH?

charle [14.2K]

Answer: FH = 16

Concept:

Bisect means to cut or divide something into two equal parts.

In mathematics, usually bisect means cutting a segment or an angle.

If you are still confused, you may refer to the attachment below for a graphical explanation or tell me.

Solve:

<u>Given information</u>

FG is 16 less than 3 times the length of GH

Let x be the length of GH

<u>Given expression</u>

FG = GH

<u>Set equation</u>

3x - 16 = x

<u>Add 16 on both sides</u>

3x - 16 + 16 = x + 16

3x = x + 16

<u>Subtract x on both sides</u>

3x - x = x + 16 - x

2x = 16

<u>Divide 2 on both sides</u>

2x / 2 = 16 / 2

x=8

--------------------------------------------------------------------------------------------------------------

Concept:

In order to find the length of FH, we need to know the idea of the segment addition postulate.

The Segment Addition Postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC.

Solve:

<u>Given information</u>

FG = GH = x

x = 8

<u>Given expression deducted from the segment addition postulate</u>

FH = FG + GH

<u>Substitute values into the expression</u>

FH = x + x

FH = 8 + 8

$\boxed {FH=16}$

Hope this helps!! :)

Please let me know if you have any questions

8 0

3 years ago

Carol is building a screen door. The height of the door is one foot more than twice its width. What is the width of the door if

Arte-miy333 [17]

Hello,

A good method for solving this question is creating an equation to solve for the width of the door.

Let w = the width of the door
Let h = the height of the door

The height (h) is twice the width (2w) and one foot more (+1).
We can make the equation h = 2w + 1

Now, we are given that the height of the door is 7 feet, so h = 7.
We can simply plug in 7 for h in the equation to solve for w.

So, we have h = 2w + 1
7 = 2w + 1
Subtract by 1 on both sides to get:
6 = 2w
Divide by 2 on both sides to get:
w = 3

The width of the door is 3 feet.

However, we should check out answer with the given question to make sure it checks out.

We are given that the height of the door is one foot more than twice its width, and the height of the door is 7 feet.

Twice the width is 6 feet, and one foot more than that is 7 feet. Our answer checks out.

The width of the door is 3 feet.

Hope this helps!

6 0

3 years ago

Which of the following is an arithmetic sequence?

Novosadov [1.4K]

The answer is B or C

7 0

3 years ago

Read 2 more answers

What is 1 4/5 + 2 3/20 +5/ 3

sveticcg [70]

$1 \frac{4}{5} +2 \frac{3}{20} + \frac{5}{3} \\ \\ 3 + \frac{4}{5} + \frac{3}{20} + \frac{5}{3} \\ \\ LCD = 60 \\ \\ 3 + \frac{4 \times 12}{5 \times 12} + \frac{3 \times 3}{20 \times 3} + \frac{5 \times 20}{3 \times 20} \\ \\ 3 + \frac{48}{60} + \frac{9}{60} + \frac{100}{60} \\ \\ 3 + \frac{48 + 9 + 100}{60} \\ \\ 3 + \frac{157}{60} \\ \\ 3 + 2 \frac{37}{60} \\ \\ 5 \frac{37}{60} \\ \\ Answer: \fbox {5 37/60} \ or \ \fbox {5.6167}$

5 0

3 years ago