Sandra Jane said:
Homework Statement: What is exactly Differential Equations?
Relevant Equations: Please use general equation examples so I can understand
So I'm new to engineering and have studied some of the calculus but until now, I still have a hard time to understand what is exactly Differential Equations, what is it for and how can I use it in the future classes as an Engineering Physics student
A differential equation is an equation that relates a quantity to how that quantity changes.
Since physics is fundamentally about change—motion, heat flow, electric fields, waves, quantum systems—differential equations become one of the main languages of physics and engineering.
Start with a simple example
Suppose you're driving a car.
- Position: x(t)x(t)x(t)
- Velocity: v(t)v(t)v(t)
From calculus:
- Velocity is the derivative of position:
v=dxdtv = $\frac{dx}{dt}v=dtdx$
If someone tells you:
dxdt=20\frac{dx}{dt}=20dtdx=20
they are saying:
"The position changes at 20 meters per second."
This is a differential equation.
The solution is:
x(t)=20t+Cx(t)=20t+Cx(t)=20t+C
where CCC is the starting position.
Notice what happened:
- The differential equation described a rule of change.
- The solution told us the actual motion.
That's the essence of differential equations.
Why physics uses them
Most physical laws describe rates of change, not final answers.
For example, Newton's Second Law:
F=maF = maF=ma
Since acceleration is the derivative of velocity,
a=d2xdt2a = \frac{d^2x}{dt^2}a=dt2d2x
Newton's law becomes:
F=md2xdt2F=m\frac{d^2x}{dt^2}F=mdt2d2x
This is a differential equation.
The equation itself is the law of nature. Solving it tells you how an object moves.
A real engineering physics example
Imagine a mass attached to a spring.
According to Robert Hooke:
F=−kxF=-kxF=−kx
Using Newton's law:
md2xdt2=−kxm\frac{d^2x}{dt^2}=-kxmdt2d2x=−kx
or
md2xdt2+kx=0m\frac{d^2x}{dt^2}+kx=0mdt2d2x+kx=0
This differential equation describes:
- springs
- vibrations
- molecules
- bridges
- electrical oscillators
Its solution is a sine/cosine wave.
So one differential equation explains many real systems.
Where you'll see differential equations in your degree
As an Engineering Physics student, you'll encounter them almost everywhere.
Mechanics
Motion of particles, rockets, pendulums, planetary orbits.
Example:
md2xdt2=F(x,t)m\frac{d^2x}{dt^2}=F(x,t)mdt2d2x=F(x,t)
Electricity and Magnetism
Maxwell's equations are differential equations.
They describe:
- electric fields
- magnetic fields
- electromagnetic waves
- radio communication
Circuits
An RC circuit satisfies:
RCdVdt+V=V0RC\frac{dV}{dt}+V=V_0RCdtdV+V=V0
This tells you how capacitors charge and discharge.
Heat Transfer
The temperature in an object changes according to differential equations.
Engineers use these to design:
- engines
- batteries
- processors
- cooling systems
Waves
Sound waves, water waves, light waves:
all come from differential equations.
Quantum Mechanics
The famous Erwin Schrödinger equation is a differential equation.
Solving it predicts:
- atomic structure
- chemical bonds
- semiconductors
- lasers
An intuition that helps
Think of differential equations as:
"Instructions for how a system changes."
An ordinary algebra equation tells you what something
is.
Example:
y=3x+2y=3x+2y=3x+2
A differential equation tells you how something
evolves.
Example:
dydx=3\frac{dy}{dx}=3dxdy=3
One is a static relationship.
The other is a rule for change.
Physics is mostly about change, so physics naturally leads to differential equations.
