KAHIBARO
Table of Contents
Why differential equations matter in physics
Physics often asks how a quantity changes. Position changes with time, temperature changes with distance, charge changes in a circuit, and populations of particles change as they decay. A differential equation is an equation that relates a function to its derivatives. In physics, this is powerful because derivatives describe rates of change.
If we know a rule for how something changes, then a differential equation can describe the whole physical process. For example, if the acceleration of an object is known, then since acceleration is the derivative of velocity, and velocity is the derivative of position, we can build equations for the motion.
A differential equation does not usually give the final answer immediately. Instead, it gives a relationship that must be solved to find the unknown function.
A differential equation is an equation involving an unknown function and one or more of its derivatives.
Functions, variables, and derivatives in equations
A differential equation usually contains an independent variable and a dependent variable. In physics, time $t$ is often the independent variable, and a quantity such as position $x(t)$ is the dependent variable.
For example,
$$
\frac{dx}{dt} = v
$$
says that the rate of change of position is velocity. If velocity is constant, then this equation leads to a straight line for position as a function of time.
Another example is radioactive decay:
$$
\frac{dN}{dt} = -\lambda N
$$
Here $N(t)$ is the number of undecayed nuclei, and $\lambda$ is a constant. The equation says that the rate of decay is proportional to how much remains.
Order of a differential equation
The order of a differential equation is the highest derivative that appears in it.
If the highest derivative is the first derivative, the equation is first order. If the highest derivative is the second derivative, it is second order.
Examples are shown below.
| Equation | Highest derivative | Order |
|---|---|---|
| $\frac{dy}{dx} = 3x$ | first | 1 |
| $\frac{dN}{dt} = -\lambda N$ | first | 1 |
| $\frac{d^2x}{dt^2} = -g$ | second | 2 |
| $\frac{d^2x}{dt^2} + \omega^2 x = 0$ | second | 2 |
In physics, many important laws involve second order equations because acceleration is a second derivative of position.
The order of a differential equation is determined by the highest derivative present.
Ordinary and partial differential equations
If a function depends on only one independent variable, such as $x(t)$, the equation is called an ordinary differential equation, or ODE.
If a function depends on several independent variables, such as temperature $T(x,t)$ depending on position and time, then derivatives with respect to more than one variable appear. This gives a partial differential equation, or PDE.
Examples:
$$
\frac{dN}{dt} = -\lambda N
$$
is an ODE, because $N$ depends only on $t$.
$$
\frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2}
$$
is a PDE, because $T$ depends on both $x$ and $t$.
At this stage, the main idea is that ODEs describe change with respect to one variable, while PDEs describe change with respect to several variables.
General solution and particular solution
When we solve a differential equation, we often first find a general solution. This contains one or more constants. These constants appear because differentiation loses information, and integration restores it only up to unknown constants.
For example, if
$$
\frac{dy}{dx} = 2x
$$
then integrating gives
$$
y = x^2 + C
$$
This is the general solution.
To determine the constant $C$, we need extra information, usually called an initial condition or a boundary condition. If we are told that $y=3$ when $x=1$, then
$$
3 = 1^2 + C
$$
so $C=2$, and the particular solution is
$$
y = x^2 + 2
$$
A general solution contains arbitrary constants. A particular solution is found after applying initial or boundary conditions.
Initial conditions and boundary conditions
In physics, the extra information needed to choose the correct solution usually comes from the physical situation.
An initial condition gives the value of the function or its derivative at a starting point, often at $t=0$. For example,
$$
x(0) = x_0, \qquad v(0) = v_0
$$
These are common in motion problems.
A boundary condition gives values at specific positions or endpoints. For example, the temperature at the ends of a rod might be fixed.
A first order equation usually needs one condition. A second order equation usually needs two conditions.
A first look at solving differential equations
Many differential equations cannot be solved by simple algebra. But some important ones in beginner physics can be solved with direct integration or separation of variables.
Direct integration
If the derivative is given directly as a function of the independent variable, we can integrate.
Example:
$$
\frac{dv}{dt} = a
$$
If $a$ is constant, then
$$
v = at + C
$$
If we know $v(0)=v_0$, then
$$
v = v_0 + at
$$
This is a familiar kinematics result.
If we then use
$$
\frac{dx}{dt} = v_0 + at
$$
and integrate again, we get
$$
x = x_0 + v_0 t + \frac{1}{2}at^2
$$
So some standard motion formulas come directly from solving differential equations.
Separation of variables
A separable equation can be rearranged so that one side contains only the dependent variable and the other side contains only the independent variable.
Consider radioactive decay:
$$
\frac{dN}{dt} = -\lambda N
$$
Rearrange:
$$
\frac{dN}{N} = -\lambda \, dt
$$
Integrate both sides:
$$
\int \frac{dN}{N} = \int -\lambda \, dt
$$
which gives
$$
\ln |N| = -\lambda t + C
$$
Exponentiating,
$$
N = N_0 e^{-\lambda t}
$$
where $N_0$ is the value at $t=0$.
This result appears throughout physics because many systems change at a rate proportional to their current amount.
If a differential equation can be written as
$$
\frac{dy}{dx} = f(x)g(y),
$$
then it may be solved by separation of variables:
$$
\frac{dy}{g(y)} = f(x)\,dx
$$
followed by integration.
Common physics forms
Some differential equations appear so often that it helps to recognize their form immediately.
Constant rate of change
$$
\frac{dy}{dt} = k
$$
This means the quantity changes at a constant rate. The solution is linear:
$$
y = kt + C
$$
Rate proportional to the quantity
$$
\frac{dy}{dt} = ky
$$
If $k>0$, the solution grows exponentially:
$$
y = Ce^{kt}
$$
If $k<0$, the solution decays exponentially:
$$
y = Ce^{kt}
$$
with decreasing magnitude.
Constant second derivative
$$
\frac{d^2x}{dt^2} = a
$$
This describes constant acceleration. The solution is quadratic in time:
$$
x = x_0 + v_0 t + \frac{1}{2}at^2
$$
Restoring force form
$$
\frac{d^2x}{dt^2} = -\omega^2 x
$$
This leads to oscillatory motion, such as simple harmonic motion. The full study of that motion belongs elsewhere, but the important point here is that the differential equation determines the kind of behavior, in this case oscillation.
Reading physical meaning from an equation
A differential equation tells you more than just how to compute a solution. It also tells you the character of the physical process.
Consider these examples.
| Differential equation | Physical meaning | Typical behavior |
|---|---|---|
| $\frac{dy}{dt} = k$ | constant rate | straight line |
| $\frac{dy}{dt} = ky$ | rate proportional to amount | exponential growth or decay |
| $\frac{d^2x}{dt^2} = 0$ | zero acceleration | constant velocity |
| $\frac{d^2x}{dt^2} = -\omega^2 x$ | acceleration toward equilibrium | oscillation |
This is one reason differential equations are central in physics. The equation itself already reveals the type of motion or change.
A graphical view
A differential equation can be viewed as giving the slope of a curve at each point. For a first order equation, knowing $\frac{dy}{dx}$ means knowing how steep the graph is at each location.
For example, if
$$
\frac{dy}{dx} = x
$$
then the slope is negative for negative $x$, zero at $x=0$, and positive for positive $x$. This already suggests a curve that bends upward, which matches the solution
$$
y = \frac{1}{2}x^2 + C
$$
The red line segments represent slopes at different points. A differential equation can be thought of as a rule that assigns such slopes.
Why constants appear in solutions
Suppose you know only that
$$
\frac{dx}{dt} = 5
$$
Then many functions satisfy this, such as
$$
x = 5t, \qquad x = 5t + 2, \qquad x = 5t - 10
$$
All of them have the same derivative. This is why integration introduces a constant. Physically, the rate of change alone does not determine the starting value.
For second order equations, two constants usually appear because two integrations are required. Physically, these often correspond to initial position and initial velocity.
Units in differential equations
A differential equation must be dimensionally consistent. Every term must have the same units.
For example,
$$
\frac{dN}{dt} = -\lambda N
$$
The left side has units of quantity per time. So $\lambda$ must have units of inverse time.
In
$$
\frac{d^2x}{dt^2} = -\omega^2 x
$$
the left side has units of acceleration, $\text{length}/\text{time}^2$. Therefore $\omega^2$ must have units of $1/\text{time}^2$, so $\omega$ has units of $1/\text{time}$.
Checking units is a simple and important way to test whether an equation makes sense.
Every term in a differential equation must have the same physical dimensions.
Simple worked examples
Example 1, constant acceleration
Suppose
$$
\frac{dv}{dt} = 3
$$
with initial condition $v(0)=2$.
Integrate:
$$
v = 3t + C
$$
Apply the condition:
$$
2 = 3(0) + C \Rightarrow C=2
$$
So
$$
v(t) = 2 + 3t
$$
Example 2, exponential decay
Suppose
$$
\frac{dQ}{dt} = -0.5Q
$$
with $Q(0)=8$.
The solution has the form
$$
Q = Ce^{-0.5t}
$$
Apply the initial condition:
$$
8 = Ce^0 \Rightarrow C=8
$$
So
$$
Q(t) = 8e^{-0.5t}
$$
Example 3, constant second derivative
Suppose
$$
\frac{d^2x}{dt^2} = -4
$$
with $v(0)=6$ and $x(0)=1$.
Integrate once:
$$
\frac{dx}{dt} = v = -4t + C_1
$$
Using $v(0)=6$ gives $C_1=6$, so
$$
v = 6 - 4t
$$
Integrate again:
$$
x = 6t - 2t^2 + C_2
$$
Using $x(0)=1$ gives $C_2=1$, so
$$
x(t) = 1 + 6t - 2t^2
$$
What beginners should remember
Differential equations are the language of change in physics. They connect physical laws to mathematical functions. Instead of giving position, temperature, or charge directly, physics often gives a rule for how these quantities change. Solving the differential equation turns that rule into a usable formula.
You do not need to master all methods at once. At the beginner level, the most important ideas are recognizing derivatives inside physical laws, understanding order, using initial conditions, and solving simple equations by integration or separation of variables.
Key ideas:
$$
\text{Differential equation} = \text{relationship involving a function and its derivatives}
$$
$$
\text{1st order} \Rightarrow \text{highest derivative is first}
$$
$$
\text{2nd order} \Rightarrow \text{highest derivative is second}
$$
$$
\text{Initial or boundary conditions select the physical solution}
$$
$$
\text{Many physics laws are differential equations in disguise}
$$
Final intuition
A formula like $x = x_0 + v_0 t + \frac{1}{2}at^2$ is useful, but a differential equation like
$$
\frac{d^2x}{dt^2} = a
$$
is deeper. It expresses the physical law directly, in terms of change. This is why differential equations appear everywhere in physics. They are not just a mathematical technique, they are one of the main ways nature is described.