Let dy dx =x^ 2 -4 ( for x R)

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

\frac{dy}{dx}=x^{2}-4 \qquad (\text{for }x\in\mathbb R).

Two physical systems are modeled by this differential equation but begin with different initial data. (a) Determine the particular solution

y=f(x)

satisfying

f(1)=3

. (b) Determine the particular solution

y=g(x)

satisfying

g(-1)=7

. (c) Show that for all

x

\mathbb R

, the difference

g(x)-f(x)

is a constant. State the value of this constant and interpret its meaning in the context of the two systems. (d) Prove that the graphs of

y=f(x)

and

y=g(x)

never intersect. Your proof should use properties of the differential equation and the results of parts (a)–(c).