DIFFERENTIAL EQUATIONS 01
What Is a Differential Equation?
A differential equation describes a relationship involving an unknown function and one or more of its derivatives.
The main idea
An ordinary algebraic equation asks for an unknown number.
For example,
\[ 2x+3=7. \]Here the unknown is the number \(x\).
A differential equation asks for an unknown function.
Here the unknown is the function \(y(x)\).
1. What is being solved?
In
\[ \frac{dy}{dx}=2x, \]we want a function whose derivative is \(2x\).
Since
\[ \frac{d}{dx}(x^2)=2x, \]one solution is
\[ y=x^2. \]But because constants disappear when differentiated,
\[ \frac{d}{dx}(x^2+C)=2x. \]So the general solution is
2. Why differential equations are useful
Many models do not tell us a quantity directly. Instead, they describe how quickly it changes.
For example,
\[ \frac{dP}{dt}=0.2P \]says that the rate of population change is proportional to the current population.
The differential equation describes the rule of change. Its solution describes how the population evolves through time.
3. Function versus derivative
Suppose
\[ y=y(x). \]Then
\[ \frac{dy}{dx} \]describes how \(y\) changes with \(x\).
A differential equation connects this rate to other quantities.
For example,
\[ \frac{dy}{dx}=y \]says that the rate of change of \(y\) is equal to the current value of \(y\).
4. A solution must satisfy the equation
Consider
\[ \frac{dy}{dx}=y. \]The function
\[ y=e^x \]is a solution because
\[ \frac{d}{dx}e^x=e^x. \]More generally,
\[ y=Ce^x \]is also a solution because
\[ \frac{d}{dx}(Ce^x)=Ce^x. \]5. Verifying a solution
Suppose we are told that
\[ y=3e^{2x} \]solves
\[ \frac{dy}{dx}=2y. \]Differentiate:
\[ \frac{dy}{dx}=6e^{2x}. \]Also,
\[ 2y=2(3e^{2x})=6e^{2x}. \]Since both sides are equal, the function satisfies the differential equation.
6. Order of a differential equation
The order is determined by the highest derivative that appears.
For example,
\[ \frac{dy}{dx}+y=0 \]is first order.
But
\[ \frac{d^2y}{dx^2}+y=0 \]is second order.
7. First-order equations
A first-order ordinary differential equation may be written generally as
It relates the first derivative of \(y\) to \(x\), \(y\), or both.
8. Second-order equations
A second-order equation contains
\[ \frac{d^2y}{dx^2}. \]For example,
\[ \frac{d^2y}{dt^2}=-ky \]can model oscillatory motion.
The second derivative may represent acceleration, curvature, or another second-order rate of change depending on the model.
9. Ordinary differential equations
An ordinary differential equation, or ODE, involves derivatives with respect to one independent variable.
For example,
\[ \frac{dy}{dt}=ky. \]There is only one independent variable: \(t\).
10. Partial differential equations
If a function depends on several independent variables, partial derivatives may appear.
For example,
\[ \frac{\partial u}{\partial t} = D\frac{\partial^2u}{\partial x^2}. \]This is a partial differential equation, or PDE.
For now, this section focuses on ODEs.
11. General and particular solutions
A general solution usually contains arbitrary constants.
For
\[ \frac{dy}{dx}=2x, \]the general solution is
\[ y=x^2+C. \]If extra information is given, such as
\[ y(0)=3, \]then
\[ 3=0^2+C, \]so
\[ C=3. \]The particular solution is
12. Initial conditions
A condition such as
\[ y(0)=3 \]is called an initial condition.
It specifies the state of the system at a chosen starting point.
The differential equation describes how the system changes. The initial condition tells us where that evolution begins.
13. A differential equation as a rule of motion
Consider
\[ \frac{dy}{dt}=F(y,t). \]At each point \((t,y)\), the equation tells us the slope
\[ \frac{dy}{dt}. \]So an ODE can be viewed as assigning a direction of movement throughout the \((t,y)\)-plane.
14. Example: exponential growth
Suppose
\[ \frac{dP}{dt}=kP, \qquad k>0. \]The larger \(P\) becomes, the larger its growth rate becomes.
The solution is
\[ P(t)=P_0e^{kt}. \]So the differential equation expresses the local rule
\[ \text{rate}=k\times\text{current amount}, \]while the solution gives the complete trajectory through time.
15. Example: decay
If
\[ \frac{dP}{dt}=-kP, \qquad k>0, \]the rate is negative and proportional to the current amount.
The solution is
\[ P(t)=P_0e^{-kt}. \]So the quantity decreases exponentially.
16. Why solving matters
The equation
\[ \frac{dy}{dt}=F(t,y) \]describes the instantaneous rule of change.
A solution
\[ y(t) \]tells us the actual value of the quantity at each time.
17. The central idea
Differential equations reverse the usual modelling question.
Instead of beginning with a known function and asking for its derivative, we begin with information about the derivative and ask:
That question is the foundation of ordinary differential equations.