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:

$\displaystyle \frac{dy}{dx} = \frac{2\sqrt y}x \implies \frac1{\sqrt y} \, dy = \frac2x \, dx$

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 :

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

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

$\boxed{y = \left(\ln|x| + 2\right)^2}$