How Differential Equations Relate to Physics Equations – Motion, Velocity, and Acceleration

How does differential equations relate to physics equations?

Consider the second-order-linear-nonhomogeneous-ODE with constant coefficients,

So how does this relate to physics? Well, if distance is $x$, then velocity is $\dot{x}=x’=v$, and acceleration is $\ddot{x}=v’=a$. Assuming this is not constant acceleration or velocity, we will have functions of time. Thus,

In physics, all of the kinematic equations of motion are differential equations that you are solving by plug and play. In a ODE course, you would solve these using ODE techniques.

In a physics course, the ‘plug and play’ approach to solutions would be something like, “Find the time the sums acceleration, velocity, and distance to zero when $a(t)=t^2$, $v(t)=t$, $x(t)=t-1$.”

Time is positive, so we omit $-1-\sqrt{2}$ and keep $t_0=-1+\sqrt{2}\approx0.41\text{ s}$.

So, at $t=0.41\text{ s}$, the summation of acceleration, velocity, and distance is equal to zero.

ODE Method to Solve the Equation. (one of many methods)

Solve for $x(t)$, provide $a(t)+v(t)+x(t)=0$. (Same as finding $y$ in $y”+y’+y=0$.)

Rewrite as $x”(t)+x'(t)+x(t)=0$. Now you use the “Constant Coefficient Method.”

The above equation (answer) models motion (distance/displacement), the derivatives are velocity (speed) and acceleration. This is how differential equations is in relation to physics.