Differential Equations: First and Second Order
Differential Equations: First and Second Order
Differential equations model dynamic systems — population growth, oscillations, circuits, cooling. A-Level Further Maths covers several solution techniques for both first-order and second-order equations.
First-Order Equations
Separable Equations
If dy/dx = f(x)g(y), separate variables:
∫ 1/g(y) dy = ∫ f(x) dx
Example: dy/dx = xy → ∫ dy/y = ∫ x dx → ln|y| = x²/2 + C → y = Ae^(x²/2)
Integrating Factor Method
For equations of the form dy/dx + P(x)y = Q(x):
1. Compute the integrating factor μ(x) = e^(∫P(x)dx)
2. Multiply through: d/dx[μy] = μQ
3. Integrate: μy = ∫μQ dx
Example: dy/dx + 2y/x = x² (for x > 0)
- P(x) = 2/x, so μ = e^(∫2/x dx) = e^(2ln x) = x²
- d/dx[x²y] = x⁴
- x²y = x⁵/5 + C
- y = x³/5 + C/x²
Exact Equations
The equation M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x.
If exact, there exists F(x,y) such that ∂F/∂x = M and ∂F/∂y = N, and the solution is F(x,y) = C.
Second-Order Linear Equations with Constant Coefficients
The general form is:
a(d²y/dx²) + b(dy/dx) + cy = f(x)
The solution has two parts: y = y_CF + y_PI
Complementary Function (y_CF)
Solve the homogeneous equation a(d²y/dx²) + b(dy/dx) + cy = 0.
The auxiliary equation is: am² + bm + c = 0
| Discriminant | Roots | Complementary function |
|---|---|---|
| b² − 4ac > 0 | m₁, m₂ real and distinct | y = Ae^(m₁x) + Be^(m₂x) |
| b² − 4ac = 0 | m repeated | y = (A + Bx)e^(mx) |
| b² − 4ac < 0 | m = p ± qi complex | y = e^(px)(A cos qx + B sin qx) |
Example: y'' − 5y' + 6y = 0
- Auxiliary: m² − 5m + 6 = (m−2)(m−3) = 0
- Roots: m = 2, m = 3
- y_CF = Ae^(2x) + Be^(3x)
Example: y'' + 4y' + 4y = 0
- Auxiliary: m² + 4m + 4 = (m+2)² = 0
- Repeated root: m = −2
- y_CF = (A + Bx)e^(−2x)
Example: y'' + 2y' + 5y = 0
- Auxiliary: m² + 2m + 5 = 0 → m = (−2 ± √(4−20))/2 = −1 ± 2i
- y_CF = e^(−x)(A cos 2x + B sin 2x)
Particular Integral (y_PI)
For the non-homogeneous equation, guess a form for y_PI based on f(x):
| f(x) | Trial y_PI | If trial is in y_CF |
|---|---|---|
| constant k | y = λ | y = λx |
| linear px + q | y = λx + μ | y = x(λx + μ) |
| e^(kx) | y = λe^(kx) | y = λxe^(kx) (or λx²e^(kx) if repeated) |
| cos kx or sin kx | y = λcos kx + μsin kx | y = x(λcos kx + μsin kx) |
| polynomial degree n | y = polynomial degree n | Multiply by x |
Key rule: If the trial PI duplicates a term in y_CF, multiply by x (or x² if still duplicating).
Example: y'' − 5y' + 6y = 2e^(3x)
y_CF = Ae^(2x) + Be^(3x) (from earlier)
Trial y_PI = λe^(3x) — but e^(3x) is in y_CF! So try y_PI = λxe^(3x).
- y_PI' = λe^(3x) + 3λxe^(3x) = λe^(3x)(1 + 3x)
- y_PI'' = λe^(3x)(6 + 9x)
- Sub in: λe^(3x)(6+9x) − 5λe^(3x)(1+3x) + 6λxe^(3x) = 2e^(3x)
- λ(6+9x−5−15x+6x) = 2
- λ(1) = 2 → λ = 2
- y_PI = 2xe^(3x)
General solution: y = Ae^(2x) + Be^(3x) + 2xe^(3x)
Applying Boundary/Initial Conditions
The general solution has two arbitrary constants (A and B). To find them, use:
- Initial conditions: y(0) = ... and y'(0) = ...
- Boundary conditions: y at two different x-values
Coupled First-Order Equations
Systems like dx/dt = ax + by, dy/dt = cx + dy can be solved by:
1. Differentiate one equation and substitute the other to get a single second-order equation
2. Solve the second-order equation for one variable
3. Back-substitute to find the other
Damped and Forced Oscillations (Physical Interpretation)
The equation y'' + 2ky' + ω²y = 0 models damped oscillation:
- Underdamped (k < ω): oscillates with decaying amplitude → complex roots → e^(−kt)(A cos qt + B sin qt)
- Critically damped (k = ω): fastest return without oscillation → repeated root → (A + Bt)e^(−kt)
- Overdamped (k > ω): slow exponential return → two real negative roots
Adding a forcing term f(t) on the right side gives forced oscillation, where y_PI represents the steady-state response.
Exam Tips
- For first-order: always check if separable first (easiest), then try integrating factor
- Write the auxiliary equation immediately for second-order constant-coefficient problems
- When your trial PI is part of y_CF, multiply by x — this is the most common error source
- For complex auxiliary roots p ± qi, the CF uses e^(px) times trig in q, not p
- Always apply initial/boundary conditions to the general solution (CF + PI), not just the CF
- State your complete general solution clearly before applying conditions