Question

In Exercises \(15-18,\) solve the differential equation. $$F^{\prime}(x)=\frac{2}{x^{3}-x}$$

Short answer

The solution of the differential equation is \(F(x) = ln\left|\frac{x+1}{x-1}\right|+C.\)

Step-by-step solution

Simplify the function

We can simplify the function by splitting the denominator as the difference of two quantities. Therefore, \(F^{\prime}(x)=\frac{2}{x^{3}-x} = \frac{2}{x(x^{2}-1)} = \frac{2}{x(x-1)(x+1)}.\)

Apply partial fraction decomposition

We'll use the sum of fractions on the simplified function to reduce it into simpler fractions that can easily be integrated. We need find A, B, C which satisfy: \(\frac{2}{x(x-1)(x+1)} = \frac{A}{x}+\frac{B}{x-1}+\frac{C}{x+1}\)

Solve for constants A, B and C

Multiplying both sides by \(x(x-1)(x+1)\) gives $2 = A(x-1)(x+1)+Bx(x+1)+Cx(x-1). Comparing coefficients or substituting convenient values of x (0, 1, -1), we find that A=0, B=-1, C=1.

Integrate

Now, we can integrate the sum of fractions: \(\int F^{\prime}(x) dx = \int \left(\frac{B}{x-1}+\frac{C}{x+1}\right) dx = -\int \frac{1}{x-1} dx + \int \frac{1}{x+1} dx. By applying the integral of \(1/x\) which is \(ln|x|,\) we get: -ln|x-1|+ln|x+1| = ln\left|\frac{x+1}{x-1}\right|.\)

Write Final Solution

To obtain \(F(x),\) add the integration constant, C, to the integral \(-ln|x-1|+ln|x+1|.\) Hence, the solution of the differential equation is \(F(x) = ln\left|\frac{x+1}{x-1}\right|+C.\)