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3 years ago

3xy'+y=12x find general solution

1 answer:

5 0

The equation is linear, as you can write

$3xy'+y=12x\implies y'+\dfrac1{3x}y=4$

An integrating factor would be

$\mu(x)=\exp\left(\displaystyle\int\frac{\mathrm dx}{3x}\right)=x^{1/3}$

Multiplying both sides of the ODE by this yields

$x^{1/3}y'+\dfrac1{3x^{2/3}}y=4x^{1/3}$

where the LHS is a derivative:

$\dfrac{\mathrm d}{\mathrm dx}\left[x^{1/3}y\right]=4x^{1/3}$

Integrating both sides, we get

$x^{1/3}y=4\displaystyle\int x^{1/3}\,\mathrm dx$
$x^{1/3}y=3x^{4/3}+C$
$y=3x^{3/3}+Cx^{-1/3}=3x+\dfrac C{x^{1/3}}$

You might be interested in

In how many ways can 7 persons be arranged in a line such that (a) two particular persons are never together.

ivanzaharov [21]

Answer:

3600 ways

Step-by-step explanation:

person A has 7 places to choose from :

→ He has 2 places ,one to the extreme left of the line ,the other to the extreme right of the line

If he chose one of those two ,person B will have 5 choices and the other 5 persons will have 5! Choices.

⇒ number of arrangements = 2×5×5! = 1 200

→ But Person A also , can choose one of the 5 places in between the two extremes .

If he chose one of those 5 ,person B will have 4 choices and the other 5 persons wil have 5! Choices.

⇒ number of arrangements = 5×4×5! = 2 400

In Total they can be arranged in :

1200 + 2400 = 3600 ways

3 0

1 year ago

A practice law exam has 100 questions, each with 5 possible choices. A student took the exam and received 13 out of 100.If the s

Cloud [144]

Answer:

$z=\frac{13-20}{4}=-1.75$

Assuming:

H0: $\mu \geq 20$

H1: $\mu$

$p_v = P(Z$

Step-by-step explanation:

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest (number of correct answers in the test), on this case we now that:

$X \sim Binom(n=100, p=0.2)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

We need to check the conditions in order to use the normal approximation.

$np=100*0.2=20 \geq 10$

$n(1-p)=20*(1-0.2)=16 \geq 10$

So we see that we satisfy the conditions and then we can apply the approximation.

If we appply the approximation the new mean and standard deviation are:

$E(X)=np=100*0.2=20$

$\sigma=\sqrt{np(1-p)}=\sqrt{100*0.2(1-0.2)}=4$

So we can approximate the random variable X like this:

$X\sim N(\mu =20, \sigma=4)$

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean". The letter $\phi(b)$ is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words: $\phi(b)=P(z$