Question

D.sqrt(2+x^/2) Solve this question please

Answer 1

Answer:

Option a.

Step-by-step explanation:

By looking at the options, we can assume that the function y(x) is something like:

$y = \sqrt{4 + a*x^2}$

$y' = (1/2)*\frac{1}{\sqrt{4 + a*x^2} }*(2*a*x) = \frac{a*x}{\sqrt{4 + a*x^2} }$

such that, y(0) = √4 = 2, as expected.

Now, we want to have:

$y' = \frac{x*y}{2 + x^2}$

replacing y' and y we get:

$\frac{a*x}{\sqrt{4 + a*x^2} } = \frac{x*\sqrt{4 + a*x^2} }{2 + x^2}$

Now we can try to solve this for "a".

$\frac{a*x}{\sqrt{4 + a*x^2} } = \frac{x*\sqrt{4 + a*x^2} }{2 + x^2}$

If we multiply both sides by y(x), we get:

$\frac{a*x}{\sqrt{4 + a*x^2} }*\sqrt{4 + a*x^2} = \frac{x*\sqrt{4 + a*x^2} }{2 + x^2}*\sqrt{4 + a*x^2}$

$a*x = \frac{x*(4 + a*x^2)}{2 + x^2}$

We can remove the x factor in both numerators if we divide both sides by x, so we get:

$a = \frac{4 + a*x^2}{2 + x^2}$

Now we just need to isolate "a"

$a*(2 + x^2) = 4 + a*x^2$

$2*a + a*x^2 = 4 + a*x^2$

Now we can subtract a*x^2 in both sides to get:

$2*a = 4\\a = 4/2 = 2$

Then the solution is:

$y = \sqrt{4 + 2*x^2}$

The correct option is option a.