A solution of the differential equation satisfies the initial condition .

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

Problem

A solution of the differential equation

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

satisfies the initial condition

y(0)=1

. (a) For

0\le x\le4

and

0\le y\le3

, describe how the magnitude of the slope indicated by the direction field changes as

y

increases while

x

is held fixed. Support your description with specific values of

y

. (b) Using only information available from the direction field (no analytic solution), sketch a qualitative graph of the solution curve for

0\le x\le4

. (c) Based on the pattern of slopes, estimate

\displaystyle \lim_{x\to\infty}y(x)

and justify your estimate in the context of the direction field. (d) Determine whether the solution curve possesses any horizontal asymptotes. Provide a justification that relies solely on the behavior of

\dfrac{dy}{dx}

as observed in the direction field.

Related theory

Browse the problem library