Consider the differential equation where is a non-zero real constant

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

Problem

Consider the differential equation

\displaystyle \frac{dy}{dx}=ky,

where

k

is a non-zero real constant. A family of functions that satisfies this differential equation is given by

y=C e^{kx},

where

C

is an arbitrary real constant. (a) Describe the mathematical role of the constant

C

in the expression

y=C e^{kx}

and explain why the set of all such functions is called a "general solution" of the differential equation. (b) Explain how imposing an initial condition of the form

y(x_0)=y_0

restricts the general solution to a single function, called a particular solution. (c) If the initial condition

y(0)=5

is imposed, determine the resulting particular solution and justify each step of your reasoning.

Related theory

Initial Value Problems
General and Particular Solutions
Solutions
Variable separable equations
Differential Equations

Browse the problem library