Show that the general solution to the differential equation: dy/dx = x/(k-y) can be written in the form x ² + (y - k)² = c.

Sure, let's solve this differential equation step by step.

1. First, we rewrite the differential equation in the form of separable variables. The given differential equation is dy/dx = x/(k-y). We can rewrite this as (k-y) dy = x dx.

2. Now, we integrate both sides of the equation. The integral of the left side, ∫(k-y) dy, is k*y - (1/2)y² + C₁ (where C₁ is the constant of integration). The integral of the right side, ∫x dx, is (1/2)x² + C₂ (where C₂ is the constant of integration).

3. Setting both integrals equal to each other, we get k*y - (1/2)y² + C₁ = (1/2)x² + C₂.

4. We can simplify this equation by multiplying through by 2 and rearranging terms to get 2k*y - y² = x² + 2(C₂ - C₁).

5. Now, we complete the square on the left side of the equation. To do this, we rewrite -y² + 2ky as -(y² - 2ky), and then add and subtract (k)² inside the parenthesis to get -(y² - 2k*y + k² - k²) = -(y - k)² + k².

6. Substituting this back into the equation, we get -(y - k)² + k² = x² + 2(C₂ - C₁).

7. Finally, we rearrange the equation to the form x² + (y - k)² = C, where C = k² - 2(C₂ - C₁) is a new constant.

So, the general solution to the differential equation dy/dx = x/(k-y) can be written in the form x² + (y - k)² = C.