The solution of the differential equation dy dx = x - y x + y , when x = 0 and y = 0 represents ....

Step-by-step solution

The given differential equation is dy dx = x - y x + y Rearranging the terms, we get: (x + y)dy = (x - y)dx x dy + y dy = x dx - y dx x dy + y dx = x dx - y dy This can be written as exact differentials: d(xy) = x dx - y dy Integrating both sides, we get: d(xy) = x dx - y dy xy = x^2 2 - y^2 2 + C x^2 - 2xy - y^2 = -2C Given that x = 0 when y = 0 , substituting these values gives: 0^2 - 2(0)(0) - 0^2 = -2C C = 0 Substituting C = 0 back into the equation, we obtain: x^2 - 2xy - y^2 = 0 This is a homogeneous equation