📋 Chapter Overview
What is a differential equation?
Any equation involving a derivative term — it describes a rate of change.
General vs. Particular Solution
General includes the unknown constant c; particular uses a boundary condition to find c.
Separation of Variables
A technique to solve differential equations by rearranging so each variable appears on only one side, then integrating.
Real-World Modelling
Convert word descriptions into differential equations, solve them, and interpret results in context.
1. General Solutions
Understanding what differential equations are and how to solve them with an unknown constant
What Is a Differential Equation?
A differential equation is simply any equation that contains at least one derivative. You've already seen them in disguise — whenever you differentiated and got an expression like dy/dx = 3x², that was a differential equation. Now we're reversing the process: we're given the derivative and asked to find the original function.
Differential equations are classified by order:
- First-order: Contains only first derivatives (e.g.,
dy/dx = 4x)
- Second-order: Contains second derivatives (e.g.,
d²y/dx² = 3)
Recognising Differential Equations
Example 1 (first-order):
dy/dx = 3t
This is first-order because it only contains the first derivative.
Example 2 (second-order):
d²y/dx² − 3 = 2x
This is second-order because it contains the second derivative d²y/dx².
What Is a General Solution?
When you integrate a derivative, you get a family of solutions — an infinite number of functions, all differing by a constant. That constant is the general solution.
Think of it this way: If you integrate dy/dx = 4x, you get y = 2x² + c. This isn't one function; it's infinitely many functions, each shifted vertically by a different value of c. That constant c is unknown unless you're given extra information (like a point the curve passes through).
Key Idea: A general solution always includes the constant of integration (usually written as c, or sometimes A if it's an exponential). The general solution represents a family of curves, not a single curve.
Finding a General Solution
Step 1: Integrate both sides
y = ∫4x dx = 2x² + c
Step 2: Write the general solution
y = 2x² + c (c is an arbitrary constant)
This represents infinitely many parabolas, all with the same shape but different vertical positions.
Second-Order General Solutions
When you differentiate twice, you need to integrate twice. This gives you two constants of integration, traditionally written as c and d.
Second-Order Example
Step 1: First integration (to get dy/dx)
dy/dx = ∫4 dx = 4x + c
Remember:
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