What is the solution to the differential equation $ \dfrac{{dy}}{{dx}} = {y^2} + 1 $ with $ y\left( 1 \right) = 0 $ ?

Hint: In the given question, we are given a differential equation. So, we have to solve the given differential equation using methods of integration. Then, we will find the value of the arbitrary constant c which is put after computing an indefinite integral using the information given to us in the question itself. Indefinite integration gives us a family of curves. The point given to us in the question helps us to determine the particular function.

Complete step by step solution:
The given question requires us to solve a differential equation $ \dfrac{{dy}}{{dx}} = {y^2} + 1 $ using methods of integration. First of all, we have to classify the type of differential equation.
Now, we observe that we can separate the variables in the differential equation given to us. So, the differential equation $ \dfrac{{dy}}{{dx}} = {y^2} + 1 $ is of variable separable form.
Now, we separate the variables. So, we get,
 $ \Rightarrow \dfrac{{dy}}{{{y^2} + 1}} = dx $
Now, integrate both sides.
 $ \Rightarrow \int {\dfrac{{dy}}{{{y^2} + 1}}} = \int {dx} $
We know that the integral of $ \dfrac{1}{{{p^2} + 1}} $ with respect to p is $ {\tan ^{ - 1}}\left( p \right) $ . So, we get,
 $ \Rightarrow {\tan ^{ - 1}}\left( y \right) = \int {dx} $
Now, we also know the power rule of integration. We can write $ 1 $ as $ {x^0} $ and integrate using the power rule of integration $ \int {{x^n}} = \left( {\dfrac{{{x^{n + 1}}}}{{n + 1}}} \right) $ . So, we get,
 $ \Rightarrow {\tan ^{ - 1}}\left( y \right) = x + c $ where c is any arbitrary constant.
Now, we are given that $ y\left( 1 \right) = 0 $ . This means that for $ x = 1 $ , we have $ y = 0 $ .
So, putting $ x = 1 $ and $ y = 0 $ in the equation, we get,
 $ \Rightarrow {\tan ^{ - 1}}\left( 0 \right) = 1 + c $
 $ \Rightarrow 0 = 1 + c $
Shifting the terms and finding the value of c, we get,
 $ \Rightarrow c = - 1 $
So, the value of c is $ \left( { - 1} \right) $ .
Therefore, the solution of the differential equation $ \dfrac{{dy}}{{dx}} = {y^2} + 1 $ is $ {\tan ^{ - 1}}\left( y \right) = x - 1 $
So, the correct answer is “ $ {\tan ^{ - 1}}\left( y \right) = x - 1 $ ”.

Note: The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant. Exact value of the arbitrary constant can be calculated only if we are given a point lying on the function.