dy/dx-tan y/(1+x)=(1+x) sec y

Question

Differential equation

Answer 1

Multiply both sides by cos(y)/(1+x):

[cos(y)/(1+x)]dy/dx-sin(y)/(1+x)²=1.

Let u=sin(y)/(1+x),

then du/dx=[(1+x)cos(y)dy/dx-sin(y)]/(1+x)²=

[cos(y)/(1+x)]dy/dx-sin(y)/(1+x)², so

du/dx=1 and u=x+C where C is the constant of integration.

Therefore, replacing u, we get:

sin(y)/(1+x)=x+C, sin(y)=(x+C)(x+1)=x²+(1+C)x+C.

y=arcsin(x²+(1+C)x+C).