Question

In Exercises \(15-18,\) solve the differential equation. $$\frac{d y}{d x}=\frac{2 x-6}{x^{2}-2 x}$$

Short answer

The solution to the differential equation is \( y = \ln |x^{2}-2x| + c \).

Step-by-step solution

Identify the function and its derivative

In \(\frac{2x-6}{x^{2}-2x}\), see that the numerator is the derivative of the denominator, i.e, \(f'(x) = 2x-6\) and \(f(x) = x^{2}-2x\). Thus, the equation is of the form \(\frac{dy}{dx} = \frac{f'(x)}{f(x)}\).

Integrate both sides of the equation

The equation \(\frac{dy}{dx} = \frac{f'(x)}{f(x)}\) integrates to \( \int dy = \int\frac{f'(x)}{f(x)}dx\) which simplifies to \( y = \int\frac{f'(x)}{f(x)}dx \).

Apply the formula for integrating a function divided by its derivative

The integral simplifies to \( y = \int\frac{f'(x)}{f(x)}dx = \ln |f(x)| + c \), where \(c\) is the constant of integration.

Substitute \(f(x)\) back in the equation

Substituting \(f(x) = x^{2}-2x\) in the equation, we get \( y = \ln |x^{2}-2x| + c \).

Solve for \(y\)

Finally, the solution to the differential equation is \( y = \ln |x^{2}-2x| + c \).