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3 years ago

Select the answer that represents the algebraic translation of the phrase below.

1 answer:

3 0

3(x - 2y), because it sais difference and that means subtraction and it also sais 3 times the difference of the product of x and 2y.

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y′′ −y = 0, x0 = 0 Seek power series solutions of the given differential equation about the given point x 0; find the recurrence

sukhopar [10]

Let

$\displaystyle y(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + \cdots$

Differentiating twice gives

$\displaystyle y'(x) = \sum_{n=1}^\infty na_nx^{n-1} = \sum_{n=0}^\infty (n+1) a_{n+1} x^n = a_1 + 2a_2x + 3a_3x^2 + \cdots$

$\displaystyle y''(x) = \sum_{n=2}^\infty n (n-1) a_nx^{n-2} = \sum_{n=0}^\infty (n+2) (n+1) a_{n+2} x^n$

When x = 0, we observe that y(0) = a₀ and y'(0) = a₁ can act as initial conditions.

Substitute these into the given differential equation:

$\displaystyle \sum_{n=0}^\infty (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^\infty a_nx^n = 0$

$\displaystyle \sum_{n=0}^\infty \bigg((n+2)(n+1) a_{n+2} - a_n\bigg) x^n = 0$

Then the coefficients in the power series solution are governed by the recurrence relation,

$\begin{cases}a_0 = y(0) \\ a_1 = y'(0) \\\\ a_{n+2} = \dfrac{a_n}{(n+2)(n+1)} & \text{for }n\ge0\end{cases}$

Since the n-th coefficient depends on the (n - 2)-th coefficient, we split n into two cases.

• If n is even, then n = 2k for some integer k ≥ 0. Then

$k=0 \implies n=0 \implies a_0 = a_0$

$k=1 \implies n=2 \implies a_2 = \dfrac{a_0}{2\cdot1}$

$k=2 \implies n=4 \implies a_4 = \dfrac{a_2}{4\cdot3} = \dfrac{a_0}{4\cdot3\cdot2\cdot1}$

$k=3 \implies n=6 \implies a_6 = \dfrac{a_4}{6\cdot5} = \dfrac{a_0}{6\cdot5\cdot4\cdot3\cdot2\cdot1}$

It should be easy enough to see that

$a_{n=2k} = \dfrac{a_0}{(2k)!}$

• If n is odd, then n = 2k + 1 for some k ≥ 0. Then

$k = 0 \implies n=1 \implies a_1 = a_1$

$k = 1 \implies n=3 \implies a_3 = \dfrac{a_1}{3\cdot2}$

$k = 2 \implies n=5 \implies a_5 = \dfrac{a_3}{5\cdot4} = \dfrac{a_1}{5\cdot4\cdot3\cdot2}$

$k=3 \implies n=7 \implies a_7=\dfrac{a_5}{7\cdot6} = \dfrac{a_1}{7\cdot6\cdot5\cdot4\cdot3\cdot2}$

so that

$a_{n=2k+1} = \dfrac{a_1}{(2k+1)!}$

So, the overall series solution is

$\displaystyle y(x) = \sum_{n=0}^\infty a_nx^n = \sum_{k=0}^\infty \left(a_{2k}x^{2k} + a_{2k+1}x^{2k+1}\right)$

$\boxed{\displaystyle y(x) = a_0 \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!} + a_1 \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}}$

4 0

3 years ago

Which statement about the angle measures is true?

sattari [20]

Answer:

Option (4)

Step-by-step explanation:

By the property of exterior angle of a triangle,

"Exterior angle of a triangle is equal to the sum of two opposite interior angles."

In the triangle ABC,

∠ACD is an exterior angle and ∠BAC and ∠ABC are the opposite interior angles.

m∠ACD = m∠BAC + m∠ABC

95° = m∠BAC + m∠ABC

Therefore, Option (4) will be the correct option.

5 0

3 years ago

Help please!!!!!!!!!!!!!!!!!

krok68 [10]

C. 0.25 because you divide 0.06 by 0.24

5 0

3 years ago

Find the equation of the line that is parallel to the line x + 5y = 10 and passes through the point (1,3).

Nataliya [291]

Answer:

Step-by-step explanation:

for two lines to be parallel they MUST have the same slope but have different Y intercepts ( Y axis crossing values)

parallel lines will never intersect each other

first I use a graphing calcuator to solve the problem

I like to use y intercept form better, but you don't have use it

I will try to solve it directly without going through the y intercept equation

x + 5y = 10 in y intercept form y = mx + b is

5y = -x + 10 divide both sides by 5

y = -x/5 + 10/5

y = -x/5 + 2

Now I need to find a parallel line that passes through the x =1 and y = 3 point recall this line will have the same slope = m = -1/5

and a different Y axis crossing point 'b' that we don't know

y = -x/5 + b y = 3 when x = 1 so slove for 'b'

3 = -1/5 + b

3 = -0.2 + b add -0.2 to both sides

3.2 = b

y = -x/5 + 3.2 this is answer in y intercept form

if you multiply both sides by 5

5y = -x + 16 add x to both sides

x + 5y = 16 answer in standard form

THE EASIER WAY TO SOLVE THE PROBLEM IS ......

x + 5y = 10 find a line parallel that passes through x = 1, y = 3

x + 5y = Z Z is an unknown value plug in x and y and solve for Z

1 + 5(3) = Z

Z = 16 so the parallel line in standard form is

x + 5y = 16

same as before, my y intercept method was correct, but not worth the effort

6 0

3 years ago

Rebecca bought a desk for $70 she refinished it and sold it for a 20% profit? How much profit did she make? What price did she r

belka [17]

Answer:

she sold it for $84 and made $14

Step-by-step explanation:

take 20% of 70 add it to 70 and bam answer

6 0

3 years ago