Mathematics

Problem

Consider the differential equation

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

where

y

is a differentiable function of

x

. (a) Derive the general solution of the differential equation and express it explicitly in the form

y=\displaystyle -\frac{1}{\arctan x + C},

where

C

is an arbitrary constant. (b) The function

y_0(x)= -\dfrac{1}{\arctan x}

is proposed as a solution. Determine whether

y_0

is a member of the family you obtained in part (a) and state whether it represents a general or particular solution. Justify your conclusion. (c) Discuss how varying the constant

C

affects the behavior of solutions near

x=0

. In your answer, describe the qualitative differences that would appear in a slope field for at least two distinct values of

C

Intuitive explanation

This is a separable differential equation—the key insight is that you can rearrange it so all $y$ terms are on one side and all $x$ terms are on the other, then integrate both sides independently. Once you separate variables into $\frac{dy}{y^2} = \frac{dx}{1+x^2}$ , you'll recognize the right side as a standard arctangent integral, and the left side as a power function. This transforms a differential equation into an algebraic relationship you can solve for $y$ .

Intuitive explanation

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