First-order and second-order differential equations

Learning objectives

Solve a first-order linear equation dy/dx + P(x)y = Q(x) by multiplying through by the integrating factor e^(∫P dx), recognising the left side as the derivative of a product and integrating

Find the complementary function of ay″ + by′ + cy = 0 from the roots of the auxiliary equation am² + bm + c = 0, giving Ae^(m₁x) + Be^(m₂x) for distinct real roots, (A + Bx)e^(mx) for a repeated root and e^(px)(A cos qx + B sin qx) for roots p ± qi

Find a particular integral for a right-hand side that is a polynomial, an exponential or a sine or cosine, and multiply the trial function by x when the usual form already appears in the complementary function

Write the general solution as the complementary function plus a particular integral, then use initial or boundary conditions to determine the arbitrary constants, substituting into the derivative as well as the solution where required

Apply a given substitution, such as y = vx or x = eᵗ, to reduce a differential equation to a standard solvable form, transforming each derivative correctly by the chain rule

Interpret the long-term behaviour of a solution as x → ∞, identifying whether oscillations grow, decay or persist, and relate the complementary function and particular integral to the transient and steady-state parts of a model