Question

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 1

Mathematically it is given that

`(dy)/(dx)=(1)/(y)\\\\ydy=dx...........(i)\\Integrating\\\\\int ydy=\int dx\\\\(y^(2))/(2)=x+c` where 'c' is a constant

equation i is the required differential equation

thus the solution becomes

`\therefore y=√(2x+c)`

Now we have to verify that
`y=-√(2x+c)` is a solution of the differential equation

Thus differentiating it with respect to 'x' we get

`(dy)/(dx)=(d(-√(2x+c)))/(dx)\\\\(dy)/(dx)=(-1)/(2√(2x+c))* 2\\\\\therefore (dy)/(dx)=(1)/(y)`

Hence verified