general solution dy/dx = x/y

Answer 1

To find the general solution to the differential equation dy/dx = x/y, we can rearrange the equation to separate variables:

dy/y = x*dx

Integrating both sides, we get:

ln|y| = (1/2)x^2 + C

Taking the exponential of both sides, we have:

|y| = e^((1/2)x^2 + C)

Since C is an arbitrary constant, we can rewrite this equation as:

y = ±e^C * e^(1/2)x^2

y = Ae^(1/2)x^2

where A is a constant that depends on the initial conditions of the problem. Therefore, the general solution to the differential equation dy/dx = x/y is y = Ae^(1/2)x^2.