Consider the differential equation dy dx = y^ 2 1+x^ 2

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the differential equation dydx=y21+x2,\frac{dy}{dx}=\frac{y^{2}}{1+x^{2}}, where yy is a differentiable function of xx. (a) Derive the general solution of the differential equation and express it explicitly in the form y=−1arctan⁡x+C,y=\displaystyle -\frac{1}{\arctan x + C}, where CC is an arbitrary constant. (b) The function y0(x)=−1arctan⁡xy_0(x)= -\dfrac{1}{\arctan x} is proposed as a solution. Determine whether y0y_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 CC affects the behavior of solutions near x=0x=0. In your answer, describe the qualitative differences that would appear in a slope field for at least two distinct values of CC.

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