Can someone show me how to do this equation I know what the

Can someone show me how to do this equation? I know what the answer is, but wolfram alpha is not showing me enough work. I want to know how to do this

dy/dx = (y 1)^2 + 0.04, y(0)=1

This is a separable DE, and the solution im looking for is

y=1/5tan(x/5)+1

Solution

dy/dx = (y 1)^2 + 0.04

=>

dy/[(y-1)^2+0.2^2] = dx

integration both sides

1/0.2 arctan( (y-1)/0.2 ) = x+c

since integration of 1/a^2+x^2 = 1/a arcttan(x/a)

=>

(y-1)/0.2 = tan(0.2(x+c))

y(0)=1

=>

1-1/0.2 = tan(0.2c)

=>

c =0

=>

y-1/0.2 = tan(0.2x)

=>

y = 0.2 tan(0.2x) + 1

= 1/5 tan(x/5) + 1