A function y = g(x) is described by some geometric property of its graph. Write

Question

A function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f(x, y) having the function g as its solution (or as one of its solutions). The slope of the graph of g at the point (x, y) is the sum of x and y. The line tangent to the graph of g at the point (x, y) inter- sects the x-axis at the point (x/2, 0). Every straight line normal to the graph of g passes through the point (0, 1). Can you guess what the graph of such a function g might look like? The graph of g is normal to every curve of the form y = x^2 + k (k is a constant) where they meet. The line tangent to the graph of g at (x, y) passes through the point (-y, x).

Explanation / Answer

Given that the function :

y = g(x)

27. Slope of the graph suppose m which is nothing but derivative of y represented by :

m = dy/dx = y' = x + y answer

29. The tangent line to g(x) at x=x0 passes through (x0,g(x0))with slope g(x0).

The normal line passes through (x0,g(x0)) and is perpendicular to the tangent line, so it has slope 1/g'(x0) (assuming g'(x0)0 ).

So the normal line line is given by y=g(x0)(xx0)/g'(x0).

All these lines pass through (0,1), so if you plug in x=0,y=1 into the previous equation, you get a differential equation (relating g(x0),g'(x0), and x0). The graph of the function might look like Circle and the function is continuous function.

So from question

(y-1) = (-dx/dy)(x - 0)

=> (y-1) = - x (dx/dy)

=> 1 - y = x ( dx/dy)

=> dy/dx = x/(1 - y) answer

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