A curve y = f(x) has the property that the slope of tangent to the curve at every point is reciprocal to the y value of the curve. Write down a differential equation whose solution is y = f(x). Then show verification that y = root (2 x+C) and y =-root (2x+C) satisfy this differential equation.

Answer:Mathematically it is given that[tex]\frac{dy}{dx}=\frac{1}{y}\\\\ydy=dx...........(i)\\Integrating\\\\\int ydy=\int dx\\\\\frac{y^{2}}{2}=x+c[/tex] where 'c' is a constantequation i is the required differential equationthus the solution becomes[tex]\therefore y=\sqrt{2x+c}[/tex]Now we have to verify that [tex]y=-\sqrt{2x+c}[/tex] is a solution of the differential equationThus differentiating it with respect to 'x' we get[tex]\frac{dy}{dx}=\frac{d(-\sqrt{2x+c})}{dx}\\\\\frac{dy}{dx}=\frac{-1}{2\sqrt{2x+c}}\times 2\\\\\therefore \frac{dy}{dx}=\frac{1}{y}[/tex]Hence verified