Mathematics

Problem

Two students analyze the differential equation

\frac{dy}{dx} = \ln(1 + x^{2})

. They record their work as follows: Student A:

y = 5 + \displaystyle \int_{0}^{x} \ln(1 + t^{2})\,dt

. Student B:

y = C + \displaystyle \int_{3}^{x} \ln(1 + t^{2})\,dt

, where

C

is a constant. a. Part 1: Write Student B’s expression for

y

entirely in terms of

x

and the constant

C

by evaluating the integral from

3

x

in a form that makes the comparison with Student A’s result clear. b. Part 2: Determine the value of

C

that makes Student A’s and Student B’s formulas represent the same curve in the

xy

-plane. c. Part 3: Verify directly that your value of

C

in part (b) yields

y(0)=5

for Student B’s formula. d. Part 4: Explain why, for any real number

k

, there is exactly one curve satisfying both the given differential equation and the condition

y(3)=k