Finding the particular solution of first degree order of DE using variables separable

Question

1.) xyy' - y^2 = 0 ; when x = 2, y = 1

2.) cosy dx = xdy ; when x = 1, y = 1

Answer 1

1) xyy' - y^2 = 0 ; when x = 2, y = 1

xyy' =y^2

y;/y =1/x

lny =lncx

y=cx

y(2)=1 ==> 1=c*2 ---> c=1/2

y=x/2

y=0 is another solution

Similar for 2)

2.) cosy dx = xdy ; when x = 1, y = 1

2.) dy/cosy= dx/x

∫dy/cosy ==∫dx/x

find ∫dy/cosy, and equate to RHS ln(cx)