Urgent help math

Question

Consider the IVP

dy/dx = y/x -αsqrt1-y/x y(a)=0.

(a) Show that this equation is homogeneous.

(b) Find the explicit solution to this IVP.

(c) Forα<1,α=1andα>1,findlimy(x). x→0

(d) For a = 200, k = 1/5 and 0 ≤ x ≤ 200, find the maximum of y (x).

Answer 1

dy/dx = y/x - a(1 - y/x)^1/2; y(a) = 0

(a) This is Homoginius equation, because right side f(x, y ) = y/x - a(1 - y/x)^1/2 = f(kx, ky)

(b) L<et y = vx, where v is new function of x. Then dy/dx = v + xdv/dx and we sabstitute this in our O)DE:

v + xdv/dx = v - a(1 - v)^1/2; dv/(1 - v)^1/2 = - adx/x. We integrate both sides: ∫dv/(1 - v)^1/2 = ∫-adx/x

-2(1 - v)^1/2 = -alnIxI - C; Let put back v = y/x; 2(1 - y/x)^1/2 = alnIxI + C. If x = a y(a) = 0; 2 = alnIaI + C;

C = 2 - alnIaI

2(1 - y/x)^1/2 = alnIxI + 2 - alnIaI; 4(1 - y/x) = (alnIx/aI + 2)²; y = x(1 - (alnIx/aI + 2)²/