Differential Equations

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ODE i.e., Ordinary Differential Equations and PDE i.e. (that is), Partial Differential Equations. You are solving ODEs and PDEs all the time—long before you even get to a Differential Equations course.

If you are in a calculus-based physics course. Chances are you don’t know calculus yet, unless you took calculus in high school or prior, but most students are doing it as a corequisite, which I highly recommend you don’t do. Sometimes it is unavoidable but if you can get through all three calculus courses prior to physics, your life will be much easier.

Let us consider a very basic ODE and PDE:

$$ y’=x \quad\Leftrightarrow\quad \frac{dy}{dx}=x \quad\Leftrightarrow\quad f'(x)=x, \qquad x’=t \quad\Leftrightarrow\quad \frac{dx}{dt}=t \quad\Leftrightarrow\quad x'(t)=t=\dot{x}. $$

The equation $y’=x$ (y prime equals x) is referred to as a first-order-linear-nonhomogeneous-ODE.

$$ \frac{ \color{green}{dy}\leftarrow[\text{dependent variable}] }{ \color{red}{dx}\leftarrow[\text{independent variable}] } =x, \qquad \frac{ \color{red}{dx}\leftarrow[\text{dependent variable}] }{ \color{green}{dy}\leftarrow[\text{independent variable}] } = \frac{1}{x} $$

The equation $x’=\frac{1}{x}$ is a first-order-homogeneous-non-linear-ODE.

Homogenize (verb)- [1] subject (milk) to a process in which the fat droplets are emulsified and the cream does not separate. [2] make uniform or similar.

Homogenization (noun) [1] a process by which the fat droplets from milk are emulsified and the cream does not separate. [2] the process of making things uniform or similar.

Homogeneous (adjective)- This has many definitions from course-to-course and in English—everyday speaking vs. mathematics. Stay tuned to your textbook and course for proper use.

It is not pronounced homo gize ness THERE IS NO z in homogeneous. It is not potato po tah toe.

Now, starting with $y’=x$ we move into $\frac{dy}{dx}=x\Rightarrow dy=x\,dx$. This is a separable variable ODE.

Now, starting with $y’=x$ we move into $\frac{\partial y}{\partial x}=x\Rightarrow dy=x\,dx$. This is a separable variable PDE.

$$ dy=x\,dx \quad\Rightarrow\quad \int dy=\int x\,dx \quad\Rightarrow\quad y+c_1=\frac{x^2}{2}+c_2 \quad\Rightarrow\quad y=\frac{x^2}{2}+c_3 = \frac{1}{2}x^2+C. $$

$$ \therefore\quad y(x)=\frac{1}{2}x^2+C. $$

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This lesson is nothing more than a free preview of The Ultimate Crash Course for STEM Majors. The complete Crash Course series goes further into ordinary differential equations, partial differential equations, calculus, physics, engineering mathematics, and advanced STEM problem solving.

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