Differential equations
Order, degree, formulating and solving simple differential equations.
What is a differential equation?
A differential equation (DE) is an equation involving an unknown function and its derivatives, such as dy/dx = 3x² or d²y/dx² + y = 0.
- Order: the order of the highest derivative present.
- Degree: the power of the highest-order derivative when the equation is a polynomial in its derivatives. Degree is not defined if derivatives appear inside functions such as sin(dy/dx).
Example: d³y/dx³ + (d²y/dx²)² + y = 0 has order 3 and degree 1.
Forming differential equations
A DE can come from a family of curves (eliminate the arbitrary constants by differentiating) or from a situation. The number of arbitrary constants equals the order. In real life, "the rate of change of a quantity is proportional to the quantity itself" gives dA/dt = kA, whose solution is A = A₀e^(kt). This models population growth, continuous compound interest (k = r) and radioactive decay (k < 0).
Example: money grows continuously at 5% per year, so dA/dt = 0.05A and A = A₀e^(0.05t). It doubles when e^(0.05t) = 2, that is, t = log 2/0.05 ≈ 13.86 years.
Solving first-order equations
Variable separable
Rearrange to g(y) dy = f(x) dx and integrate both sides. Example: dy/dx = x/y gives y dy = x dx, so y² = x² + C, i.e. y² − x² = C.
Linear equations
dy/dx + Py = Q (P, Q functions of x). Multiply by the integrating factor IF = e^(∫P dx); then y × IF = ∫Q × IF dx + C. Example: dy/dx + 2y = 4 has IF = e^(2x), so ye^(2x) = 2e^(2x) + C, giving y = 2 + Ce^(−2x).
Particular solutions
Use the given condition to find C. Example: dy/dx = 3x² with y(1) = 2 gives y = x³ + C and 2 = 1 + C, so y = x³ + 1.
Key points
- Order = highest derivative; degree = its power (if polynomial in derivatives).
- General solution has as many constants as the order.
- dA/dt = kA ⇒ A = A₀e^(kt).
- Linear DE: IF = e^(∫P dx).
Exam tips
- Before finding degree, make sure the equation is free of radicals and fractional powers of derivatives.
- In growth questions, use log 2 for doubling and log 4 = 2 log 2 for quadrupling.