Which is a factor of each term of the polynomial? <br><br> (3d2

Consider the following differential equation. x^2y' + xy = 3 (a) Show that every member of the family of functions y = (3ln(x) +

Veronika [31]

Answer:

Verified

$y(x) = \frac{3Ln(x) + 3}{x}$

$y(x) = \frac{3Ln(x) + 3 - 3Ln(3)}{x}$

Step-by-step explanation:

Question:-

- We are given the following non-homogeneous ODE as follows:

$x^2y' +xy = 3$

- A general solution to the above ODE is also given as:

$y = \frac{3Ln(x) + C }{x}$

- We are to prove that every member of the family of curves defined by the above given function ( y ) is indeed a solution to the given ODE.

Solution:-

- To determine the validity of the solution we will first compute the first derivative of the given function ( y ) as follows. Apply the quotient rule.

$y' = \frac{\frac{d}{dx}( 3Ln(x) + C ) . x - ( 3Ln(x) + C ) . \frac{d}{dx} (x) }{x^2} \\\\y' = \frac{\frac{3}{x}.x - ( 3Ln(x) + C ).(1)}{x^2} \\\\y' = - \frac{3Ln(x) + C - 3}{x^2}$

- Now we will plug in the evaluated first derivative ( y' ) and function ( y ) into the given ODE and prove that right hand side is equal to the left hand side of the equality as follows:

$-\frac{3Ln(x) + C - 3}{x^2}.x^2 + \frac{3Ln(x) + C}{x}.x = 3\\\\-3Ln(x) - C + 3 + 3Ln(x) + C= 3\\\\3 = 3$

- The equality holds true for all values of " C "; hence, the function ( y ) is the general solution to the given ODE.

- To determine the complete solution subjected to the initial conditions y (1) = 3. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:

$y( 1 ) = \frac{3Ln(1) + C }{1} = 3\\\\0 + C = 3, C = 3$

- Therefore, the complete solution to the given ODE can be expressed as:

$y ( x ) = \frac{3Ln(x) + 3 }{x}$

- To determine the complete solution subjected to the initial conditions y (3) = 1. We would need the evaluate the value of constant ( C ) such that the solution ( y ) is satisfied as follows:

$y(3) = \frac{3Ln(3) + C}{3} = 1\\\\y(3) = 3Ln(3) + C = 3\\\\C = 3 - 3Ln(3)$

- Therefore, the complete solution to the given ODE can be expressed as:

$y(x) = \frac{3Ln(x) + 3 - 3Ln(3)}{y}$

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6 0

2 years ago

HELP ME IN 10 MINUTES!!!! ASAP!!!!!!!!!!!! THX! NO FILES! <br>Will give brainliest!

kotegsom [21]

Answer:

Step-by-step explanation:

whats the question?

7 0

2 years ago

Determine

Luden [163]

Answer:

$\mu_{x} = 64, \sigma_{x} = 12.83$

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sample means with size n of at least 30 can be approximated to a normal distribution with mean $\mu_{x} = \mu$ and standard deviation $\sigma_{x} = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 64, \sigma = 77, n = 36$

So

$\mu_{x} = 64, \sigma_{x} = \frac{77}{\sqrt{36}} = 12.83$

7 0

3 years ago

Everything i need to know

denis23 [38]

Answer:

whales have a big pp and there poop is for a lot of money

Step-by-step explanation:

8 0

2 years ago

Read 2 more answers

Davis Electronics manufactures a certain type of radio that requires 18 working resistors per radio. If one out of 22 resistors

REY [17]

Since One Radio in Davis Electronics requires 18 working resistors, then 2,145 radios will require $18\times2145=38610$ resistors.

Now we know that one out of 22 resistors is defective. This means that the number of non defective or perfect resistors in a set of 22 resistors is 21.

So, obviously, if we pick a set of 40,370 resistors then the set of working resistors we will get is $40370\times \frac{21}{22} =38535$ .

As can be clearly seen from the above calculations the required amount of working resistors is 38610 and the amount of working calculators available to us is 38535.

Thus, since the required amount of working resistors is greater than the amount of working resistors available, 40,370 resistors are not enough to assemble 2,145 radios.

7 0

2 years ago

Which is a factor of each term of the polynomial? (3d2 – 10d)