Consider the new differential equation y dx/x^2 + y^2 - x dy/x^2 + y^2 = 0. Is t

Question

Consider the new differential equation y dx/x^2 + y^2 - x dy/x^2 + y^2 = 0. Is the equation separable? No justification required. Why is the equation homogeneous? Rewrite it in a form y' = g(x, y) that makes obvious that the equation is homogeneous. Use the method for solving homogeneous equations to solve the equation of part (b). Show all details of the calculation and give all the solutions of the y' = g(x, y) together with their domains. Describe geometrically the graphs of the solutions. Find the set S of points (x_0, y_0) for which the existence theorem for differential equations does not guarantee a solution of the initial value problem y' = g(x, y) with y(x_0) = y_0? Specify the set S; no justification required. What are all the solutions of the original differential equation. Specify all the solutions together with their domains. Describe geometrically the graphs of the solutions. No justification required. Is it true that the graph of any two solutions of the original differential equation intersect? If so, what are the possible points of intersection? No justification required.

Explanation / Answer

The give quation is ydx /(x2 + y2) - xdy / (x2 + y2) = 0

Implies ydx /(x2 + y2) = xdy / (x2 + y2)

Implies ydx = xdy

Implies dy / y = dx / x

Integrating both sides, we get

logy = log x + c

Or y/x = A

Or y = Ax.

So, the questions given follows from the above.

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