Consider again the family of solutions to .

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

Problem

Consider again the family of solutions

y=C e^{k x}

\dfrac{dy}{dx}=k y

. (i) Explain the difference between a general solution and a particular solution of a differential equation, illustrating your answer with

y=C e^{k x}

. (ii) Describe, without calculation, how the shape of the graph of

y=C e^{k x}

changes when: (a)

k

is positive, (b)

k

is negative. Comment on the roles played by

k

and

C

. (iii) A student claims that the graph of

y=C e^{k x}

can cross the

x

-axis for some choice of

C

. Do you agree? Justify your answer.