Thomas wants to put a fence around a rectangular garden that is 12 feet long and 7 feet wide. How much fencing will he need?

Find all the solutions for the equation:

Contact [7]

$2y^2\,\mathrm dx-(x+y)^2\,\mathrm dy=0$

Divide both sides by $x^2\,\mathrm dx$ to get

$2\left(\dfrac yx\right)^2-\left(1+\dfrac yx\right)^2\dfrac{\mathrm dy}{\mathrm dx}=0$

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2\left(\frac yx\right)^2}{\left(1+\frac yx\right)^2}$

Substitute $v(x)=\dfrac{y(x)}x$ , so that $\dfrac{\mathrm dv(x)}{\mathrm dx}=\dfrac{x\frac{\mathrm dy(x)}{\mathrm dx}-y(x)}{x^2}$ . Then

$x\dfrac{\mathrm dv}{\mathrm dx}+v=\dfrac{2v^2}{(1+v)^2}$

$x\dfrac{\mathrm dv}{\mathrm dx}=\dfrac{2v^2-v(1+v)^2}{(1+v)^2}$

$x\dfrac{\mathrm dv}{\mathrm dx}=-\dfrac{v(1+v^2)}{(1+v)^2}$

The remaining ODE is separable. Separating the variables gives

$\dfrac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=-\dfrac{\mathrm dx}x$

Integrate both sides. On the left, split up the integrand into partial fractions.

$\dfrac{(1+v)^2}{v(1+v^2)}=\dfrac{v^2+2v+1}{v(v^2+1)}=\dfrac av+\dfrac{bv+c}{v^2+1}$

$\implies v^2+2v+1=a(v^2+1)+(bv+c)v$

$\implies v^2+2v+1=(a+b)v^2+cv+a$

$\implies a=1,b=0,c=2$

Then

$\displaystyle\int\frac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=\int\left(\frac1v+\frac2{v^2+1}\right)\,\mathrm dv=\ln|v|+2\tan^{-1}v$

On the right, we have

$\displaystyle-\int\frac{\mathrm dx}x=-\ln|x|+C$

Solving for $v(x)$ explicitly is unlikely to succeed, so we leave the solution in implicit form,

$\ln|v(x)|+2\tan^{-1}v(x)=-\ln|x|+C$

and finally solve in terms of $y(x)$ by replacing $v(x)=\dfrac{y(x)}x$ :

$\ln\left|\frac{y(x)}x\right|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C$

$\ln|y(x)|-\ln|x|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C$

$\boxed{\ln|y(x)|+2\tan^{-1}\dfrac{y(x)}x=C}$

7 0

3 years ago

40 POINTS!!!!!!!! NEED HELP

aniked [119]

Where is the grid are u gonna take a pic?

6 0

2 years ago

A student is making a project that requires a large triangular base. The student has two pieces of wood that are 20in and 17in l

IRISSAK [1]

Answer:

Write a short story about a character who experiences a turning point in life. Use this Writing Plan/Graphic Organizer below to format (construct) your story. It must also be submitted with your short story and I will issue you two separate grades (one for the organizer one for the write-up).

Start with a writing plan (organizer)

1. Title - Your choice

2. Audience - Classmates and teacher

3. Theme - Your choice

4. Purpose - To entertain, share insight

5. Time Frame - Start today and due on 11-7-20 at 11:59 pm

6. Beginning - Introduce characters etc.

7. Middle - Turning point

8. End - Resolution

Use a minimum 150 words

Step-by-step explanation:

Write a short story about a character who experiences a turning point in life. Use this Writing Plan/Graphic Organizer below to format (construct) your story. It must also be submitted with your short story and I will issue you two separate grades (one for the organizer one for the write-up).

Start with a writing plan (organizer)

1. Title - Your choice

2. Audience - Classmates and teacher

3. Theme - Your choice

4. Purpose - To entertain, share insight

5. Time Frame - Start today and due on 11-7-20 at 11:59 pm

6. Beginning - Introduce characters etc.

7. Middle - Turning point

8. End - Resolution

Use a minimum 150 words

7 0

3 years ago

What is 4(2(329)) LIKE i have no idea how to do that

boyakko [2]

I think it wants you to multiply the numbers.
329×2=658
658×4=2632

Final Answer=2632

5 0

3 years ago

Read 2 more answers

A state trooper temp track of the number of speeding ticket he gave out each week over the course of seven weeks in the table be

NikAS [45]

Answer:

B: The center of data is shown by the mean, which is 43.

Step-by-step explanation:

Add up all the digits (33+38+38+40+44+51+57)/7=43 the mean. The mean is always teh center of data so it's B. Hope this helps.

3 0

3 years ago