Let satisfy the differential equation dy dx = k x+1 x(x-3)

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Let

y=f(x)

satisfy the differential equation

\frac{dy}{dx}=\frac{k x+1}{x(x-3)},

where

k

is a real constant. Additional data are

f(1)=0

and

f(6)=2

. (a) Solve the differential equation to obtain a formula for

f(x)

in terms of

k

and

x

. (b) Use the two data points to determine the value of

k

. (c) With this value of

k

, evaluate

\displaystyle \int_{1}^{6}\!f'(x)\,dx

and interpret the result in terms of the Fundamental Theorem of Calculus.