Differential Equation Modeling

Question

Quick disclaimer:

1. This is not graded homework - all homework is assigned but not turned in.
2. There is no assigned book, and hence no answers to given problems.
3. These questions are for the purpose of studying for a quiz.

That said, here are the problems:

A function $y=g(x)$ is described by some geometric property of its graph. Write a differential equation of the form $\frac{dy}{dx} = f(x,y)$, having $g(x)$ as its solution.

Question 1) The slope of the graph of g at the point (x,y) is the sum of x and y

Question 2) Every straight line normal to the graph of g passes through the point (0,1)

Answer 1

The first one is simple, you just read off the answer from the problem statement. I will leave it to you.

The second one is a little bit harder. The tangent line to $g(x)$ at $x=x_0$ passes through $(x_0,g(x_0))$ with slope $g'(x_0)$. The normal line passes through $(x_0,g(x_0))$ and is perpendicular to the tangent line, so it has slope $-\frac{1}{g'(x_0)}$ (assuming $g'(x_0) \neq 0$ of course). So the normal line line is given by $y=g(x_0)-\frac{x-x_0}{g'(x_0)}$. 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 $g(x_0),g'(x_0)$, and $x_0$).

Answer 2

here is another way to look at question (b). look at the tangent at $(x,y).$ since the normal goes through $(0,1),$ it has slope $\frac{y-1}x.$ therefore the tangent at $(x, y)$ has slope $-\frac{y-1}x.$ we can write the differential equations as $$\frac{dy}{dx} = -\frac{y-1}x \to y-1 = \frac Cx\to y = 1+ \frac Cx. $$