Consider the differential equation dy dx =e^ x^ 2 ( for all x R)

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

Problem

Consider the differential equation

\frac{dy}{dx}=e^{x^{2}}\quad (\text{for all }x\in\mathbb R).

A family of antiderivative curves arises from this equation when different initial conditions are imposed at the same point

x=a

. Throughout this problem, assume

a

is a fixed real number. (a) Derive an explicit expression for the particular solution

y=F_{y_{0}}(x)

satisfying

F_{y_{0}}(a)=y_{0}

, where

y_{0}

is an arbitrary real number. (b) Describe, using precise mathematical reasoning, how changing the value of

y_{0}

affects the graph of

y=F_{y_{0}}(x)

. In your explanation, address both the shape of the curve and its position relative to other members of the family. (c) Suppose two points

P(0,2)

and

Q(0,5)

lie on curves in this family. Determine whether

P

and

Q

can lie on the same particular solution. Justify your conclusion rigorously, citing any relevant theorems or properties of solutions to first-order differential equations.

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