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bearhunter [10]
3 years ago
8

Find a solution to the initial value problem, y′′+18x=0,y(0)=5,y′(0)=1.

Mathematics
1 answer:
Serga [27]3 years ago
7 0

We want to find a solution to the initial value problem:

y'' + 18x = 0 \qquad,\qquad y(0) = 5 \qquad,\qquad y'(0)=1.

We can start by integrating the equation once:

\dfrac{\textrm{d}^2 y}{\textrm{d}x^2} + 18 x = 0 \iff \dfrac{\textrm{d}^2 y}{\textrm{d}x^2} = -18 x \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -18\displaystyle\int x\textrm{ d}x \iff \dfrac{\textrm{d}y}{\textrm{d}x}=-18\dfrac{x^2}{2} + C \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + C.

Using the initial condition y'(0) = 1, we can determine the integration constant C:

\dfrac{\textrm{d}y}{\textrm{d}x}\Big\vert_{x= 0} = 1 \iff -9 \times 0^2 + C = 1 \iff C = 1.

Therefore, we have:

\dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + 1

We can now integrate again:

y(x) = \displaystyle\int\dfrac{\textrm{d}y}{\textrm{d}x}\textrm{ d}x = \int\left(-9x^2+1\right)\textrm{d}x = -9\int x^2\textrm{ d}x + \int\textrm{d}x =\\\\= -9\dfrac{x^3}{3} + x + K = -3x^3 + x + K.

The integration constant K is determined by using y(0) = 5:

y(0) = 5 \iff -3 \times 0^3 + 0 + K = 5 \iff K = 5.

Finally, the solution is:

\boxed{y(x) = -3x^3 + x + 5}.

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Please Hurry ...Which expression is equivalent to
swat32

Answer:

\huge\boxed{\sf \frac{160rs^5}{t^6}}

Step-by-step explanation:

\sf 5r^6t^4 ( \frac{4r^3s^tt^4}{2r^4st^6} ) ^5

Using rule of exponents \sf a^m/a^n = a^{m-n}

\sf 5r^6t^4 ( 2 r^{3-4} s^{2-1}t^{4-6})^5\\5r^6t^4(2r^{-1}st^{-2})^5\\5r^6t^4 * 32 r^{-5}s^5t^{-10}

Using rule of exponents \sf a^m*a^n = a^{m+n}

\sf 160 r^{6-5}s^5t^{4-10}

\sf 160 rs^5 t^{-6}

To equalize the negative sign, we'll move t to the denominator

\sf \frac{160rs^5}{t^6}

8 0
2 years ago
ABC and EDC are straight lines. EA is parallel to DB. EC = 8.1 cm. DC = 5.4 cm. DB = 2.6 cm. (a) Work out the length of AE. cm (
harkovskaia [24]

By applying the knowledge of similar triangles, the lengths of AE and AB are:

a. \mathbf{AE = 3.9 $ cm}\\\\

b. \mathbf{AB = 2.05 $ cm} \\\\

<em>See the image in the attachment for the referred diagram.</em>

<em />

  • The two triangles, triangle AEC and triangle BDC are similar triangles.
  • Therefore, the ratio of the corresponding sides of triangles AEC and BDC will be the same.

<em>This implies that</em>:

  • AC/BC = EC/DC = AE/DB

<em><u>Given:</u></em>

EC = 8.1 $ cm\\\\DC = 5.4 $ cm\\\\DB = 2.6 cm\\\\AC = 6.15 $ cm

<u>a. </u><u>Find the length of </u><u>AE</u><u>:</u>

EC/DC = AE/DB

  • Plug in the values

\frac{8.1}{5.4} = \frac{AE}{2.6}

  • Cross multiply

5.4 \times AE = 8.1 \times 2.6\\\\5.4 \times AE = 21.06

  • Divide both sides by 5.4

AE = \frac{21.06}{5.4} = 3.9 $ cm

<u>b. </u><u>Find the length of </u><u>AB:</u>

AB = AC - BC

AC = 6.15 cm

To find BC, use AC/BC = EC/DC.

  • Plug in the values

\frac{6.15}{BC} = \frac{8.1}{5.4}

  • Cross multiply

BC \times 8.1 = 6.15 \times 5.4\\\\BC = \frac{6.15 \times 5.4}{8.1} \\\\BC = 4.1

  • Thus:

AB = AC - BC

  • Substitute

AB = 6.15 - 4.1\\\\AB = 2.05 $ cm

Therefore, by applying the knowledge of similar triangles, the lengths of AE and AB are:

a. \mathbf{AE = 3.9 $ cm}\\\\

b. \mathbf{AB = 2.05 $ cm} \\\\

Learn more here:

brainly.com/question/14327552

3 0
2 years ago
Triangle ABC has coordinates A(-2, -3), B(1, 1), and C(2, -1). If the triangle is translated 4 units up, what are the coordinate
DochEvi [55]

Answer:

The coordinates of point A' will be: A'(-2, 0)

Step-by-step explanation:

We know that a translation is basically a transformation that occurs when an object is moved from one place to another place.

Translation does not change the size, orientation, or shape of the original object.

Translating a point P(x, y) 4 units up means we need to add to use P'(x, y+4), that is adding 4 units to the y-coordinate of the point P'(x, y).

Now we have to translate the triangle 4 units up. So, use the formula

P(x, y) → P'(x, y + 3)

Thus, the coordinates of point A' will be:

A(-2, -3) → A'(-2, -3+3) → A'(-2, 0)

Therefore, the coordinates of point A' will be: A'(-2, 0)

8 0
3 years ago
Mrs.smith earns $3500 a month. She saves $280 a month. Savings is what percent of her income?
arlik [135]

Answer:

3500 280

Step-by-step explanation:

280/3500

280 divided by 3500 times 100 (type this into a calculator)

its 8 her savings are 8 percent of her income

4 0
3 years ago
10 pt
lord [1]

Answer:

C. 434π

Step-by-step explanation:

Given:

Radius (r) = 7 in.

Height (h) = 24 in.

Required:

Surface area of the cylinder

Solution:

S.A = 2πrh + 2πr²

Plug in the values

S.A = 2*π*7*24 + 2*π*7²

S.A = 336π + 98π

S.A = 434π

5 0
3 years ago
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