A differential equation is given by dy dx =x ,y^ 2

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

Problem

A differential equation is given by

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

A family of functions is defined by

y(x)=\frac{1}{C-x^{2}},

where

C

is a real parameter and

C\neq x^{2}

on the domain of the function. (a) Compute

\dfrac{dy}{dx}

for the given

y(x)

. (b) Show algebraically that the function

y(x)=\dfrac{1}{C-x^{2}}

satisfies the differential equation for all real values of the parameter

C

on its domain. (c) The particular solution through the point

(0,2)

is sought. Determine the value of

C

and express this particular solution explicitly. (d) State the largest open interval containing

x=0

on which your particular solution in part (c) is valid and explain your reasoning.