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

[y/(2 + y^2)] dy = dx/[x(1 + x^2)]

Integral [y/(2 + y^2)] dy = Integral dx/[x(1 + x^2)]

(1/2) Ln (2 + y^2) = Integral dx/[x(1 + x^2)]

1/[x(1 + x^2)] = 1/x - x/(1 + x^2)

(1/2) Ln (2 + y^2) = Integral dx/[x(1 + x^2)] = Ln x - (1/2) Ln (1 + x^2) + c

(1/2) Ln (2 + y^2) = (1/2) [2 Ln x - Ln (1 + x^2)] + c

Ln (2 + y^2) = 2 Ln x - Ln (1 + x^2) + C

Ln (2 + y^2) = Ln [x^2 /(1 + x^2)] + C

e^[Ln (2 + y^2)] = e^[Ln [x^2 /(1 + x^2)] + C]

2 + y^2 = A x^2 /(1 + x^2)

y(1) = 0

2 + 0 = A 1/(1 + 1)

2 = A/2

A = 4

y^2 = 4x^2/(1 + x^2) - 2

y = sqrt(4x^2/(1 + x^2) - 2)

b) dy/dx + y/x = cos 2x ; y(PI) = 0

x dy/dx + y = x cos 2x

(d/dx) (xy) = x cos 2x

xy = integral (x cos 2x) dx

xy = (1/2) x sin (2x) - (1/2) Integral (sin 2x) dx

xy = (1/2) x sin 2x + (1/4) cos 2x + c

y(pi) = 0

0 = (1/2) pi * 0 + (1/4) (1) + c

c = -1/4

y = (1/2) sin 2x + (1/(4x)) cos 2x - 1/(4x)

d) dy/dx - y cotx = tan ^2 x ; y(PI/4) = 0

I think that - y cot x should be + y cot x

sin x dy/dx + y cos x = (sin x)^3 / (cos x)^2

(d/dx) (y sin x) = (1 - (cos x)^2) sin x / (cos x)^2

y sin x = Integral (sin x/ (cos x)^2 - sin x) dx

y sin x = 1/(cos x) + cos x + c

y(pi/4) = 0

0 = 1/(sqrt(2)/2) + sqrt(2)/2 + c

0 = sqrt(2) + sqrt(2)/2 + c

0 = (3/2) sqrt(2) + c

c = -(3/2) sqrt(2)

y = 1/( sin x cos x) + cot x - (3/2) sqrt(2) /sin x

y = csc x sec x + cot x - (3/2) sqrt(2) csc x

<_< nobody won't answer!

9! = ?