Find all functions f having the indicated property: All tangents to the graph of f pass through the origin

Equation of a tangent line to the graph of function  at point  is given by 

Since every tangent passes through the origin  we have

Let us write the equation using usual notation for differential equations.

Now we separate the variables.

Integrating the equation, we get

 is just some constant so  is also some constant. It is only more convenient to write it this way.

Taking antilogarithm gives...

Equation of a tangent line to the graph of function  at point  is given by 

Since every tangent passes through the origin  we have

Let us write the equation using usual notation for differential equations.

Now we separate the variables.

Integrating the equation, we get

 is just some constant so  is also some constant. It is only more convenient to write it this way.

Taking antilogarithm gives us the final result.

There fore, our functions  have form  where  

Graphically speaking these are all the lines that pass through the origin. Since the tangent to a line at any point is the line itself the required property is fulfilled.

Graph of several such functions  can be seen in the picture below.