The differential equation
has characteristic equation
<em>r</em> ⁴ - <em>n </em>² <em>r</em> ² = <em>r</em> ² (<em>r</em> ² - <em>n </em>²) = <em>r</em> ² (<em>r</em> - <em>n</em>) (<em>r</em> + <em>n</em>) = 0
with roots <em>r</em> = 0 (multiplicity 2), <em>r</em> = -1, and <em>r</em> = 1, so the characteristic solution is
For the non-homogeneous equation, reduce the order by substituting <em>u(x)</em> = <em>y''(x)</em>, so that <em>u''(x)</em> is the 4th derivative of <em>y</em>, and
Solve for <em>u</em> by using the method of variation of parameters. Note that the characteristic equation now only admits the two exponential solutions found earlier; I denote them by <em>u₁ </em>and <em>u₂</em>. Now we look for a particular solution of the form
where
where <em>W</em> (<em>u₁</em>, <em>u₂</em>) is the Wronskian of <em>u₁ </em>and <em>u₂</em>. We have
and so
So we have
and hence
Finally, integrate both sides twice to solve for <em>y</em> :
The area around the given curve according to the green theorem is .
According to the statement
we have to find the area enclosed by the simple closed curve that encloses the origin.
So, We know that the
The given equation is
and
If function is in form of,
and C is any positively oriented simple closed curve that encloses the origin.
Then,by use of Green's theorem
Do the partial differentiation of the given function
Then
and
On substitution in Green's theorem,
We get the value
From this it is clear that the area around the given curve is zero.
So, The area around the given curve according to the green theorem is .
Learn more about Green theorem here
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2x+y=6
y=6-2x and y=x^2-4x+3
The solution occurs when they are equal so we can say y=y which is:
x^2-4x+3=6-2x add 2x to both sides
x^2-2x+3=6 subtract 6 from both sides
x^2-2x-3=0 factor
(x+1)(x-3)=0
So x=-1 and 3, using y=6-2x we find the corresponding y values...
y(-1)=6--2=8, y(3)=6-6=0 so the two solutions are the points:
(-1,8) and (3,0)
Answer:
25pi
Step-by-step explanation:
To solve this, create a right triangle between the two points. We know the distance of one leg is 6, and the distance of the other leg is 8. Use the Pythagorean theorem to solve for the diameter. You substitute 6 and 8 into the values of A and B. Once you get this, you get (6^2)+(8^2)=c^2. Once you further simplify this, you get 36 + 64 = c^2, which is 100 = c^2. Take the square roots of both sides to get the diameter as 10. Divide the diameter by 2 to get the radius. Square the radius and multiply it by pi to get the area. Doing this will get the result of 25 pi.