Differential Equation (Solution known, but I don't understand)

[negatifzeo]

Homework Statement

In this problem, a function y= g(x) is described. Write a differential equation of the form "dy/dx=f(x,y) having the function g as it's solution (or one of it's solutions).

The line tangent to the graph of g at the point (x,y) intersects the x-axis at the point (x/2,0).

Homework Equations

The Attempt at a Solution

The solution is y'=2y/x. I'm having trouble seeing both why and how.

 

[Dick]

The tangent passes through a point on the curve, call it (x,y) and has a slope y'. That tangent also passes through (x/2,0). Express y' by finding the slope of the line through the two points (x,y) and (x/2,0).

 

[negatifzeo]

Thank you so much!

 

[Altabeh]

Homework Statement

In this problem, a function y= g(x) is described. Write a differential equation of the form "dy/dx=f(x,y) having the function g as it's solution (or one of it's solutions).

The line tangent to the graph of g at the point (x,y) intersects the x-axis at the point (x/2,0).

Homework Equations

The Attempt at a Solution

The solution is y'=2y/x. I'm having trouble seeing both why and how.

Write the equation of line with m = g'(x), x₀=x/2 and y₀=0. Then try to recover that ODE.

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