Mathematics

Problem

Consider the differential equation

\frac{dy}{dx}=2y

and the function

y=5e^{2x}

. (a) Verify that

y=5e^{2x}

satisfies the differential equation for all real

x

. Show all supporting work. (b) The general solution to the differential equation can be written in the form

y=Ce^{2x}

. Determine the value of the constant

C

that guarantees the particular solution passes through the point

(0,5)

, and justify your result. (c) Use your work from parts (a) and (b) to explain why no other function of the form

y=Ce^{2x}

can pass through the point

(0,5)

while still satisfying the differential equation.

Intuitive explanation

This problem tests whether you understand exponential growth and can verify a solution by substitution. The key insight is that exponential functions have a special property: their derivative is proportional to themselves. To check if $y=5e^{2x}$ solves the equation, compute $\frac{dy}{dx}$ directly and see if it equals $2y$ —this is how you verify any proposed solution to a differential equation.

Intuitive explanation

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