Question

form the differential equation of which the given function is a solution x2+y2+2gx+2fy+c

Answer 1

Form the differential equation of which the given function is a solution $x^{2} + y^{2} + 2gx + 2fy + c$.

**Solution:**

$$u(y)dy = v(x)dx$$ $$u(y) = 2y + 2f$$ $$v(x) = -2x - 2g$$ $$(2y + 2f)dy = -(2x + 2g)dx$$ $$\int (2y + 2f)dy = -\int (2x + 2g)dx$$ $$y^{2} + 2fy = -x^{2} - 2gx - c,$$

where $c = \text{const}$.

So, the differential equation is:

$$(2y + 2f)dy = -(2x + 2g)dx$$ $$y' = -\frac{x + g}{y + v}$$

**Answer:** $y' = -\frac{x + g}{y + v}$.