Differential Equations

Differential equations

A differential equation relates a function to its derivatives (rates of change). Solving one means finding the original function. They model real-world change: population growth, cooling, motion, mixing.

First-order: separation of variables

If you can write the equation as dy/dx = f(x)g(y), separate the variables (all y on one side, all x on the other) and integrate both sides:

∫ (1/g(y)) dy = ∫ f(x) dx

Include a constant of integration; use a boundary condition to find it (a particular solution). Without a condition you get the general solution.

First-order linear: the integrating factor

For an equation of the form dy/dx + P(x)y = Q(x):

1. Find the integrating factor I = e^(∫P dx).

2. Multiply through by I; the left side becomes d/dx(Iy).

3. Integrate both sides: Iy = ∫IQ dx, then solve for y.

Second-order (constant coefficients)

For a(d²y/dx²) + b(dy/dx) + cy = 0, form the auxiliary equation am² + bm + c = 0:

• Two real roots m₁, m₂ → y = Ae^(m₁x) + Be^(m₂x).
• Repeated root m → y = (A + Bx)e^(mx).
• Complex roots p ± qi → y = e^(px)(A cos qx + B sin qx).

For a non-zero right-hand side, add a particular integral to the complementary function.

Modelling

• Exponential growth/decay: dy/dt = ky → y = Ae^(kt).
• Simple harmonic motion: d²x/dt² = −ω²x → oscillating solution.

Boundary/initial conditions give the specific model.

Worked example

Solve dy/dx = 2xy by separation of variables.

• Separate: (1/y) dy = 2x dx.
• Integrate: ln|y| = x² + c.
• So y = Ae^(x²) (where A = e^c). ✓

Common mistakes

• Forgetting the constant of integration (and not using a boundary condition).
• Not fully separating the variables before integrating.
• For second-order, forgetting the particular integral when the RHS is non-zero.

Exam tips

• Recognise the type: separable, linear (integrating factor), or second-order (auxiliary equation).
• Always apply boundary/initial conditions for a particular solution.
• Interpret the solution in the context of the model.

Key facts to remember

• Separation of variables: rearrange to ∫(1/g(y))dy = ∫f(x)dx (+ constant, use a condition).
• Integrating factor I = e^(∫P dx) for dy/dx + Py = Q → d/dx(Iy) = IQ.
• Second-order via the auxiliary equation am² + bm + c = 0 (real/repeated/complex roots give different solution forms).