Finding a Solution to a Differential Equation with Complex Solutions

[Jameson]

Find a solution to the following D.E.

\frac{dy}{dx} + \frac{x}{y}=0

\frac{dy}{dx}=-\frac{x}{y}

Separate variables...

ydy = -xdx

Integrate both sides...

\frac{y^2}{2}=-\frac{x^2}{2}

Multiply both sides by 2, and here is where my problem arises...

y^2=-x^2

Stuck. x^2 will always be positive, so after applying the negative, I can't take the squareroot. It has to be a simple mistake. Please give a small bit of help or a small hint.

 

[Dr Avalanchez]

Aaargh... nearly there...

y^2=-x^2+C

Does this ring a bell?

 

[Integral]

Is there some reason you must restrict yourself to the reals? Even with the constant of integration which you need (as above) there is the possibility of a complex solution. A complete problem statement will include the boundary conditions or initial values. You have not provided a complete problem statement. Without that. your solution is complete with the addion of the constant of integration.