Question

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

Answer 1

Let y = f(x) be the equation of the curve, x₀ be the abscissa of the point with the fixed ordinate, x be the abscissa of the point with the variable ordinate. According to the conditions we have an equation:

$$\int_{x_0}^{x} f(t)dt = C(f(x)-f({x_0})),$$

where C is the constant of proportionality. Let's differentiate this equation with respect to x and we will have differential equation

$$f(x)=C f'(x).$$

Solution of this differential equation is $f(x)={C_1} exp(\frac{x}{C})$, where C₁ is an arbitrary real constant.

So we have a set of curves which satisfy the equation

$$y = {C_1} exp(\frac{x}{C})$$