Differential equations

What differential equations covers

Equations whose unknown is a function rather than a number. Integrating factors handle the linear first order case, auxiliary equations handle the second order one, and the same solutions describe oscillation, damping and coupled systems. Core Pure content, examinable on both compulsory papers of 9FM0, and the unit that carries the most modelling language in Core Pure.

The main ideas

• Recognising the linear first order form, and rearranging an equation into it so the coefficient of the derivative is 1.
• The integrating factor, which collapses the left-hand side into a single derivative ready to integrate.
• A general solution as a family of curves, with a boundary condition selecting one member, and sketches showing at least two.
• Second order equations with constant coefficients, through the auxiliary quadratic and its three root cases.
• Non-homogeneous equations as a complementary function plus a particular integral, with the trial multiplied by x where it clashes.
• Simple harmonic motion and damped oscillation read off the equation, with light, critical and heavy damping classified by the discriminant.
• Coupled first order systems reduced to a single second order equation by differentiating one and substituting the other.

The results it turns on

dy/dx + P(x)y = Q(x), with integrating factor e^{∫P dx}

the linear first order routine

am² + bm + c = 0

the auxiliary equation, whose roots classify the solution

distinct real roots give Ae^m₁x + Be^m₂x; a repeated root gives (A + Bx)e^mx

the two real cases

complex roots p ± qi give e^px(A cos qx + B sin qx)

the oscillating case, at angular frequency q inside a decaying envelope

general solution = complementary function + particular integral

an equation with a right-hand side

d²x/dt² = −ω²x, with period 2π/ω

simple harmonic motion, read without solving anything

Where it usually goes wrong

• The integrating factor formula assumes the coefficient of the derivative is 1, so divide through first. No constant is needed inside the exponential, since any choice cancels.
• The repeated-root case carries the factor (A + Bx). Dropping the x is the standard error in the second order lesson.
• Boundary conditions go into the full general solution, complementary function and particular integral together, and a condition on the derivative needs that solution differentiated before substituting.
• If the trial particular integral already solves the homogeneous equation, multiply it by x before continuing.

Where to start

First order equations and integrating factors first. Second order equations next, and give the three root cases separate practice, since the modelling lesson reads all three back as physical behaviour. Modelling last. Integration by parts and the standard integrals are assumed throughout.