Consider the differential equation dy dx =2y and the function .

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

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.