Find the equation of the curve passing through the origin if the middle point of the segment of its normal from any point of the curve to the x-axis lies on the parabola `2y^2=x`.

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Find the equation of the circle, passing through the origin and the foci of the parabolas y^(2) = 8x and x^(2) = 24 y .

Obtain the equation of the curve passing through the point(4,3) and at any point the gradient of the tangent to the curve is the reciprocal of the ordinate of the point.

Find the equation of the curve which passes through the point (4, 3) and the gradient of the tengent to the curve at any point on it is equal to the reciprocal of the ordinate of the point.

Find the equation of a curve passing through the point (0,2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

Find the equation of a curve passing through the point (0,1).If the slope of the tangent to the curve at any point (x,y) is equal to the sum of the x coordinate(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

Find the equation of a curve passing through the point (-2,3), given that the slope of the tangent to the curve at any point (x,y) is (2x)/(y^(2)) .

Find the equation to the curve passing through (2,2) and having its slope (-(x+y)/x) at the point (x,y)

Find the equation of a curve passing through the point (0,0) and whose differential equation is y' = e^(x) sin x .

Find the equation of the normal to curve x^(2) = 4y which passes through the point (1, 2).