Under the USMCA agreement, exports will increase to both Canada&Mexico

Solve the given initial-value problem. x^2y'' + xy' + y = 0, y(1) = 1, y'(1) = 8

Kitty [74]

Substitute $z=\ln x$ , so that

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dz}\cdot\dfrac{\mathrm dz}{\mathrm dx}=\dfrac1x\dfrac{\mathrm dy}{\mathrm dz}$

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1x\dfrac{\mathrm dy}{\mathrm dz}\right]=-\dfrac1{x^2}\dfrac{\mathrm dy}{\mathrm dz}+\dfrac1x\left(\dfrac1x\dfrac{\mathrm d^2y}{\mathrm dz^2}\right)=\dfrac1{x^2}\left(\dfrac{\mathrm d^2y}{\mathrm dz^2}-\dfrac{\mathrm dy}{\mathrm dz}\right)$

Then the ODE becomes

$x^2\dfrac{\mathrm d^2y}{\mathrm dx^2}+x\dfrac{\mathrm dy}{\mathrm dx}+y=0\implies\left(\dfrac{\mathrm d^2y}{\mathrm dz^2}-\dfrac{\mathrm dy}{\mathrm dz}\right)+\dfrac{\mathrm dy}{\mathrm dz}+y=0$
$\implies\dfrac{\mathrm d^2y}{\mathrm dz^2}+y=0$

which has the characteristic equation $r^2+1=0$ with roots at $r=\pm i$ . This means the characteristic solution for $y(z)$ is

$y_C(z)=C_1\cos z+C_2\sin z$

and in terms of $y(x)$ , this is

$y_C(x)=C_1\cos(\ln x)+C_2\sin(\ln x)$

From the given initial conditions, we find

$y(1)=1\implies 1=C_1\cos0+C_2\sin0\implies C_1=1$
$y'(1)=8\implies 8=-C_1\dfrac{\sin0}1+C_2\dfrac{\cos0}1\implies C_2=8$

so the particular solution to the IVP is

$y(x)=\cos(\ln x)+8\sin(\ln x)$

4 0

3 years ago

What is the unit rate of $46 for 5 toys

natita [175]

$46 divided by 5 which equals $9.50

8 0

3 years ago

Read 2 more answers

Please help i dont understand thank you

Alenkasestr [34]

Answer:

Part a)

We need to find the equation of a straight line passing through two given points in slope-intercept form

Part b)

The information given; we are given two points where the line passes through; (0, -4) and (-2, 2)

Part c)

We shall first determine the slope of the line using the formula;

change in y/change in x. Next, we determine the value of the y-intercept using the general form of the equation of a straight line in slope-intercept form; y = mx+c

Part d)

The slope of the line is calculated as;

(2--4)/(-2-0) =6/-2 = -3

The equation of the line in slope-intercept form becomes;

y = -3x +c

We use the point (0, -4) to determine the value of c;

-4 = -3(0)+c

c = -4

Part e)

Final solution thus becomes;

y=-3x-4

8 0

3 years ago

Jack likes to go fishing. While waiting for the fishes to bite, he formulates the following model for the process: fishes bite a

gogolik [260]

Answer:

a) 0.122

b) 3.35 × 10⁻⁴

c) 0.1071

Step-by-step explanation:

Here we have the Poisson process given by

$P(N(t) = k) = \frac{e^{-\lambda t} (\lambda t)^k }{k!}$

Where:

λ = Rate = 4 bites/hour and

t = Length of time that is

Mean = λt

Where k = 6 fishes bites and t = 2 hours we have

$P(N(2) = 6) = \frac{e^{-4 \times 2} (4 \times 2)^6 }{6!} = 0.122$

b) The probability that he fails to catch any fishes during the first two hours is the probability t that he catches 0 fish as follows

$P(N(2) = 0) = \frac{e^{-4 \times 2} (4 \times 2)^0 }{0!} = 3.35 \times 10^{-4}$

(c) Here we have

$P(N(2) = 6) = 0.122$

For 1 fish caught in two times we have first time no fish and second time he caches a fish gives

0.122×(1-0.122) = 0.1071

4 0

3 years ago

Which one?? help please for a grade!! due today

dimulka [17.4K]

Answer:

its 10

Step-by-step explanation:

because all you have to do is add a zero for each place you go

6 0

3 years ago

Read 2 more answers