Question

PLS HELP

This question is designed to be answered without a calculator.
The solution of dy/dx = (2sqrt(y))/x passing through the point (-1,4) is y =
1. ln^2|x|+2
2. ln^2|x|+4
3. (ln^2|x|+2)^2
4. (ln^2|x|+4)^2

Answer 1

The given differential equation is separable:

`\displaystyle (dy)/(dx) = \frac{2\sqrt y}x \implies \frac1{\sqrt y} \, dy = \frac2x \, dx`

Integrate both sides:

`\displaystyle \int y^(-1/2) \, dy = \int \frac2x \, dx`

`2y^(1/2) = 2\ln|x| + C`

`2\sqrt y = 2 \ln|x| + C`

Given that y(-1) = 4, we find

`2\sqrt4 = 2\ln|-1| + C \implies C = 4 - 2\ln(1) = 4`

Then

`2\sqrt y = 2 \ln|x| + 4`

Solve for y :

`\sqrt y = \ln|x| + 2`

`\left(\sqrt y\right)^2 = \left(\ln|x| + 2\right)^2`

`\boxed + 2\right)^2`