Solution:

To solve the given differential equation:
\(\frac{{(x^2 - 1)}}{{y^2}} \cdot \frac{{dy}}{{dx}} - 5(x^2 - 1) = \frac{{4x}}{{y}}\)

Let's simplify the equation step by step.

Step 1: Multiply both sides of the equation by \(y^2\) to eliminate the denominator:
\((x^2 - 1) \cdot \frac{{dy}}{{dx}} \cdot y^2 - 5(x^2 - 1) \cdot y^2 = 4x\)

Step 2: Expand the terms:
\((x^2 - 1) \cdot y^2 \cdot \frac{{dy}}{{dx}} - 5x^2y^2 + 5y^2 = 4x\)

Step 3: Group the terms with \(\frac{{dy}}{{dx}}\) together:
\((x^2 - 1) \cdot y^2 \cdot \frac{{dy}}{{dx}} = 4x + 5x^2y^2 - 5y^2\)

Step 4: Divide both sides of the equation by \((x^2 - 1) \cdot y^2\) to isolate \(\frac{{dy}}{{dx}}\):
\(\frac{{dy}}{{dx}} = \frac{{4x + 5x^2y^2 - 5y^2}}{{(x^2 - 1) \cdot y^2}}\)

Step 5: Rewrite the equation in a simplified form:
\(\frac{{dy}}{{dx}} = \frac{{4x}}{{(x^2 - 1) \cdot y^2}} + \frac{{5x^2y^2 - 5y^2}}{{(x^2 - 1) \cdot y^2}}\)

Step 6: Factor out \(5y^2\) from the second term in the numerator:
\(\frac{{dy}}{{dx}} = \frac{{4x}}{{(x^2 - 1) \cdot y^2}} + \frac{{5y^2(x^2 - 1)}}{{(x^2 - 1) \cdot y^2}}\)

Step 7: Cancel out the common factors of \((x^2 - 1) \cdot y^2\) in the numerator and denominator:
\(\frac{{dy}}{{dx}} = \frac{{4x}}{{(x^2 - 1) \cdot y^2}} + 5\)

Hence, the solution to the given differential equation is \(\frac{{dy}}{{dx}} = \frac{{4x}}{{(x^2 - 1) \cdot y^2}} + 5\).