Question

Show that the solution of the differential equation dy/dx=1+xy²+x+y²,y(0)=0 is y=tan(x+x²/2).

Answer 1

Answer:

The solution is

$\bf{y=tan(x+\frac{x^2}{2})}$

Step-by-step explanation:

Show that the solution of the differential equation dy/dx=1+xy²+x+y²,y(0)=0 is y=tan(x+x²/2).

I have applied variable separable to solve this problem

$\frac{dy}{dx}=1+xy^2+x+y^2$

$\frac{dy}{dx}=1(1+x)+y^2(1+x)$

$\frac{dy}{dx}=(1+y^2)(1+x)$

separating the variables, we get

$\frac{dy}{1+y^2}=(1+x)\:dx$

$\int\frac{dy}{1+y^2}=\int(1+x)\:dx$

$tan^{-1}y=x+\frac{x^2}{2}+c$.....(1)

But when x=0, y=0

$\implies\:tan^{-1}0=0+\frac{0^2}{2}+c$

$\implies\:c=0$

(1) becomes

$tan^{-1}y=x+\frac{x^2}{2}$

$\implies\:\boxed{y=tan(x+\frac{x^2}{2})}$