Cant figure out these Differential Equations...?

Question

1) Solve the differential equation 2xy' = sqrt(25 - y^2)

satisfying the initial condition y(1/2) = 0

2) Solve the differential equation

dy/dx = (2x^2 - 4y^2) / (xy)

Answer 1

1) This DE is separable:

dy/√(25 - y²) = dx/(2x)

Integrate both sides:

arcsin(y/5) = ln(2x) + C

Using the initial condition,

arcsin(0) = ln(1) + C

0 = C

arcsin(y/5) = ln(2x)

y = 5 sin(ln(2x))

2)

dy/dx = (2x² - 4y²) / (xy) = 2x/y - 4y/x

Observe that this has the form

dy/dx = f(y/x)

so this DE is homogeneous. Let u = y/x; then

y = xu

dy/dx = x du/dx + u

and the DE becomes

x du/dx + u = 2/u - 4u

xu du/dx = 2 - 5u²

Again, a separable equation:

u du/(2 - 5u²) = dx/x

Integrating,

(-1/5)ln|2 - 5u²| = ln|x| + C

Replacing u with y/x,

(-1/5)ln|2 - 5(y/x)²| = ln|x| + C