Differential Equations

A differential equation relates a function to its derivatives. At A-Level you focus on first-order differential equations that can be solved by separating variables.

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What Is a Differential Equation?

A differential equation contains derivatives such as dy/dx, d²y/dx², etc.

Order = the highest derivative present. A first-order ODE involves only dy/dx.

Examples

Equation	Order	Type
dy/dx = 3x²	1	Directly integrable
dy/dx = 2y	1	Separable
dy/dx = x/y	1	Separable
d²y/dx² + 4y = 0	2	Beyond A-Level Pure

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Separation of Variables

If a first-order ODE can be written as:

dy/dx = f(x) g(y)

then we can separate and integrate:

∫(1/g(y)) dy = ∫f(x) dx

Worked Example 1

Solve dy/dx = 3x²y.

Separate: (1/y) dy = 3x² dx

Integrate: ln|y| = x³ + c

So: y = Ae^(x³) where A = ±e^c.

Worked Example 2

Solve dy/dx = (x + 1)/y², given y = 2 when x = 0.

Separate: y² dy = (x + 1) dx

Integrate: y³/3 = x²/2 + x + c