Question

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy dx = f(Ax + By + C)

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

Answer 1

$\frac{dy}{dx}=(x+y+1)²$

Let (x+y+1) = t

Differentiating with respect to x ,

1+ $\frac{dy}{dx} = \frac{dt}{dx}$

=> $\frac{dy}{dx} = \frac{dt}{dx}-1$

So the differential equation becomes

$\frac{dt}{dx}-1 = t²$

=> $\frac{dt}{dx} = (1+t²)$

=> $\frac{dt}{1+t²} = dx$

Integrating ,

$\int \frac{dt}{1+t²} = \int dx$

=> $tan^{-1}t = x +$ C , where C is integration constant.

=> tan$^{-1}$ ( x + y + 1 ) = x + C

So the solution of the given differential equation is tan$^{-1}$ (x + y + 1) = x + C where C is integration constant.