Question

Solve the differential equation. \(\frac{d y}{d x}=\frac{1}{(x-1) \sqrt{-4 x^{2}+8 x-1}}\)

Short answer

The solution to the differential equation is \( y = \frac{1}{\sqrt{3}} \arctan(\frac{2(x - 1)}{\sqrt{3}}) + C \)

Step-by-step solution

Prepare the Integral for Solution

Firstly, rewrite the equation as \( \int dy = \int \frac{dx}{(x-1) \sqrt{-4 x^{2}+8 x-1}} \)

Simplify the Integral

The right-hand side of the equation can be written as the sum of perfect squares by completing the square. Simplify it by keeping the terms inside the square root in the form of (ax+b)²: \( -4 x^{2}+8 x-1 = -4(x - 1)^2 +3 \)

Integrate the Simplified Equation

The equation can now be rearranged for easier integration: \( \int = \int \frac{dx}{(x-1) \sqrt{4(x - 1)^2 -3 }} \). Using the variable substitution method \(u = x -1 \), we get the simplified form: \( \int \frac{du}{u \sqrt{4u^2 -3 }} \). This can be evaluated by using the standard integral form, which yields the answer: \( \frac{1}{\sqrt{3}} \arctan(\frac{2u}{\sqrt{3}}) + C \)

Substitute the Variable Back Into the Expression

Finally replace the variable 'u' by 'x - 1' to get the solution: \( y = \frac{1}{\sqrt{3}} \arctan(\frac{2(x - 1)}{\sqrt{3}}) + C \)