- #1

zenterix

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- Homework Statement
- My question is about some notes from MIT OCW's 18.03 "Differential Equations" on first-order ODE autonomous systems.

- Relevant Equations
- See below.

Here are the notes.

We have the system

$$\begin{bmatrix} x'\\y' \end{bmatrix}=\begin{bmatrix}f(x,y)\\g(x,y)\end{bmatrix}\tag{1}$$

We eliminate ##t## by dividing one equation by the other

$$\frac{y'}{x'}=\frac{dy/dt}{dx/dt}=\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\tag{2}$$

$$\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\tag{3}$$

which is a single first order equation involving ##y## as a function of ##x##.

I am not seeing where ##F(x,y)=0## comes from.

We have the system

$$\begin{bmatrix} x'\\y' \end{bmatrix}=\begin{bmatrix}f(x,y)\\g(x,y)\end{bmatrix}\tag{1}$$

We eliminate ##t## by dividing one equation by the other

$$\frac{y'}{x'}=\frac{dy/dt}{dx/dt}=\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\tag{2}$$

$$\frac{dy}{dx}=\frac{g(x,y)}{f(x,y)}\tag{3}$$

which is a single first order equation involving ##y## as a function of ##x##.

Indeed, in the older literature, little distinction was made between the system and the single equation - "solving" meant to solve either one.

There is however a difference between them: the system involves time, whereas the single ODE does not. Consider how their respective solutions are related:

$$\begin{matrix} x=x(t)\\y=y(t)\end{matrix}\implies F(x,y)=0\tag{4}$$

where the equation on the right is the result of eliminating ##t## from the pair of equations on the left.

**My first question**is about (4).I am not seeing where ##F(x,y)=0## comes from.

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