Solving Math Problems (Part 3)

These are math problems presented in Erwin Kreyszig's Advanced Engineering Mathematics, 6th Edition.  We have been working on the problems at the end of section 1.2

Section 1.2, Problem 10

Solve

(y - b) y' = a - x

Since the variables are already separated, we can proceed immediately to integrate both sides:

This is basically the equation for a circle centered at (a,b).  Some algebraic manipulation (multiply through by 2, moving everything over to one side, complete the squares by adding a²+b² and further rearrangement gets you to a standard circle equation.

(x - a)² + (y - b)² = R² = 2C + a² + b²

Section 1.2, Problem 11

Solve

Separating the variables we get:

Integrating both sides:

Rearranging gives the seemingly impossible:

So I had to run this graph in Desmos and see what it looked like.

The equation creates graphs only when C < 0.  For values C ≥ 0, the function produces complex numbers.