Mathematics

Problem

A function

y

satisfies the differential equation

\frac{dy}{dx}=5x^{4}-2\sqrt{x}, \qquad x\ge 0.25,

with the initial condition

y(0.25)=-1

. (a) Determine an explicit formula for

y(x)

valid for

x\ge 0.25

. Express your answer in exact form and combine like terms. (b) Using your formula from part (a), calculate the exact value of

y(1)