Answer to Question #148443 in Differential Equations for frake

Question

The area bounded by a curve, the x-axis, a fixed ordinate, and a variable ordinate is directly proportional
to the difference between the ordinates. Find the equation of the curve. Step by step and graph

Answer 1

Area of enclosed by the ordinates x=a, x=b, y=f(x) curve and x-axis

= $\int_a^bf(x)dx$

In the given problem one ordinate is fixed at (a,0) and other ordinate is at variable position (t,0)

So area will be $\int_a^tf(x)dx$

shown in attached figure.

By the problem

$\int_a^tf(x)dx = C(t-a)$ , C = constant

Differentiating both sides

$\frac{d}{dt} \int_a^tf(x)dx = \frac{d}{dt} [C(t-a)]$

=> f(t) = C

Replacing t by x we get

f(x) = C, C is a constant

Therefore equation of curve is y = C, i.e. y = constant