Find the particular solution of the differential equation: `(1+e^(2x))dy+(1+y^2)e^x dx=0,` given that `y=1,\ ` when `x=0.`

Explanation

`(1+e^(2x))dy+(1+y^(2))e^(x)dx=0`
or `(dy)/(1+y^(3))+(e^(x)dx)/(1+e^(2x))=0`
Integrating both sides, we get
`int(dy)/(1+y^(2))+int(e^(x)dx)/(1+e^(2x))=C`
`Now, tan^(-1)y+tan^(-1)(e^(x))=C`…………..(1)
`therefore tan^(-1)+tan^(-1)=C`
or `C=pi/2`
Substituting `C=pi/2`, in equation(1), we get
`tan^(-1)y+tan^(-1)(e^(x))=pi/2`
This is the required solution of the given differential equation.