Differential equations · Year 13

In Year 13 you learn: separating variables.

A differential equation links a quantity to its rate of change. Think of a population growing in proportion to its size, or tea cooling faster when it is hotter. Solving one means finding the function itself. A starting value usually fixes the constant.

Key facts

• dy/dx = f(x): integrate directly. Then use a known point to find the constant.
• Separable dy/dx = f(x)g(y): rearrange to ∫ 1/(g(y)) dy = ∫ f(x) dx.
• dy/dt = ky gives y = y₀e^(kt). This is growth if k > 0 and decay if k < 0. The half-life is (ln 2)/(|k|).
• Newton’s law of cooling: dT/dt = −k(T − R), where R is the room temperature. It gives T = R + (T₀ − R)e^(−kt).
• Linear first order dy/dx + P(x)y = Q(x): multiply through by the integrating factor e^(∫P dx).
• ay″ + by′ + cy = 0: solve am² + bm + c = 0. Distinct roots give Ae^(m₁x) + Be^(m₂x). A repeated root gives (A + Bx)e^(mx). Roots p ± qi give e^(px)(A cos qx + B sin qx).