Problem 26 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The solution to the given differential equation is \(y = 2\sqrt{x^{2}-8x+1} + C\).

Step 1: Separating the variables

First, the differential equation needs to be separated into expressions containing x and y: \( dy = \frac{x-4}{\sqrt{x^{2}-8x+1}} dx \)

Step 2: Integration

Now, both sides can be integrated: \(\displaystyle\int dy = \int \frac{x-4}{\sqrt{x^{2}-8x+1}} dx\). The left-hand side is straightforward, evaluating to y. But the right-hand side requires using a substitution method (with \(u = x^{2} - 8x + 1\)) to resolve the integral.

Step 3: Completing the integral

After substituting \(u = x^{2} - 8x + 1\), the integral on the right-hand side simplifies to \(\displaystyle\int \frac{x-4}{\sqrt{u}} du\). This integral can be computed resulting in \(2\sqrt{u} + C\). You then replace the \(u\) value with \(x^2 - 8x + 1\) to get the final solution.

Step 4: Final solution

Substitute \(u\) back into the equation to get the final solution: \(y = 2\sqrt{x^{2}-8x+1} + C\).