1
Read the rates of sine and cosine from a rounded table
State sin′ = cos and cos′ = −sin as the claims the quotients demonstrate, and conclude that the rate of the rate of cos is −cos.
Course · Differential Equations
Second-Order Linear Equations
A mass on a spring, a pendulum at small angles and a circuit with an inductor all obey the same equation: the rate of the rate of the unknown, a multiple of its rate, and a multiple of the unknown itself add up to zero. Separable Equations, Growth and Decay solved y′ = k·y with e^(kt); this course tries the same exponential on the second-order equation and finds that it works, provided r is a root of a quadratic. The quadratic is the one from Algebra, and its three possible discriminants give three shapes of solution: two exponentials, an exponential with a factor of t, and an exponential multiplied by a cosine and a sine. Every closed form is checked by substituting it into the equation, and the two constants are fitted from the starting value and the starting rate. The last two lessons look at the same equation from outside, as a pair of first-order equations whose matrix carries the quadratic inside its trace and determinant, and show Euler's method pushing a circle's worth of motion steadily outward.
a·y″ + b·y′ + c·y = 0 two constants
y = e^(rt) ⟹ a·r² + b·r + c = 0
two real roots: C₁·e^(r₁t) + C₂·e^(r₂t)
one root twice: (C₁ + C₂·t)·e^(rt)
α ± βi: e^(αt)·(C₁·cos(βt) + C₂·sin(βt))
y(0) and y′(0) fix C₁ and C₂
x′ = v, v′ = −(c/a)·x − (b/a)·v
Euler on x′ = v, v′ = −x: × (1 + h²)
What this course is about
- Subject
- Differential Equations
- Lessons
- 10
- Format
- Interactive
Overview
What you will be able to do
By the end you can turn a second-order linear equation with constant coefficients into its characteristic quadratic, solve it for every kind of root, fit both constants from starting values, check each answer exactly, and say what changes when the same equation is written as a system.
2
Verify a candidate solution of a second-order equation
Differentiate it twice, substitute, and read the residual; say why a combination C·cos t + D·sin t needs two initial values.
3
Derive and solve the characteristic equation
Substitute y = e^(rt) into a·y″ + b·y′ + c·y = 0, obtain a·r² + b·r + c = 0, and solve it exactly, with the discriminant naming the kind of root.
4
Write the general solution for each kind of root
Use two exponentials for distinct real roots, (C₁ + C₂·t)·e^(rt) for a repeated root and a damped cosine and sine for a complex pair, and read growth, decay or oscillation off the roots.
5
Fit the constants and say when the fit is guaranteed
Solve the two-by-two system from y(0) and y′(0) exactly, and use the Wronskian at 0 to say when two solutions can fit any starting values.
6
Rewrite the equation as a system and step it
Set x′ = v, find the trace and determinant that reproduce the characteristic equation, and compute the exact growth of Euler's method on x′ = v, v′ = −x.
Lessons
10 lessons
The course opens with the two functions that make the simplest second-order equation true, then asks why there are two constants. The characteristic equation is the centre: one derivation, then the three kinds of root in turn. Fitting the constants and the Wronskian say how far the answers can be trusted, and the last two lessons change the point of view from one equation to a system.
Sine, Cosine and Their RatesRead the quotients of sin and cos off a rounded table, state sin′ = cos and cos′ = −sin as the claims they demonstrate, and conclude cos″ = −cos.Not marked
The Second-Order EquationVerify that cos t, sin t and any combination solve y″ + y = 0 by exact substitution, and explain why two initial values are needed.Not marked
The Characteristic EquationSubstitute y = e^(rt) into a·y″ + b·y′ + c·y = 0, obtain a·r² + b·r + c = 0, and solve it exactly.Not marked
Real Distinct RootsWrite y = C₁·e^(r₁t) + C₂·e^(r₂t), verify it exactly, and read growth or decay off the signs of the roots.Not marked
Repeated RootsShow that one root gives one solution, verify t·e^(rt) as the second by exact substitution, and fit both constants.Not marked
Complex Roots and OscillationTurn the roots α ± βi into e^(αt)·(C₁·cos(βt) + C₂·sin(βt)), verify it exactly, and read the oscillation speed and the decay rate off the root.Not marked
Fitting the Initial ConditionsSet up the two linear equations for C₁ and C₂ from y(0) and y′(0), solve them exactly, and say when the lab declines and why that is a limit of its arithmetic and not of the method.Not marked
Superposition and the WronskianShow that a combination of solutions is a solution, compute the Wronskian at 0 exactly, and say what a nonzero value guarantees about fitting starting values.Not marked
From Second Order to a SystemRewrite a·x″ + b·x′ + c·x = 0 as x′ = v and v′ = −(c/a)·x − (b/a)·v, and show that the matrix's trace and determinant reproduce the characteristic equation.Not marked
Euler on an OscillatorStep x′ = v, v′ = −x with Euler's method, show that every step multiplies x² + v² by exactly 1 + h², and compute the growth over a whole run.Not marked
Recommended background
The previous course, First-Order Linear Equations, for the habit of checking a solution by substituting it, together with Separable Equations, Growth and Decay for the exponential solution of y′ = k·y, Differential Equations and Euler's Method for Euler's steps as exact fractions, the product rule from Rates of Change and the Derivative, and the quadratic formula and complex numbers from Algebra's Quadratics and Complex Numbers.
Practice suggestions
Substitute before you trust. Every closed form on this course is checked by differentiating it twice and putting it into the equation, and the lab prints the residual, which is exactly zero or it is not. A wrong sign in an exponent or a missing factor of t shows up there at once and does not show up in a plot.
Do the quadratic by hand first. Read a, b and c, compute b² − 4ac, and say which kind of root to expect before you pick a preset. The lab then confirms or corrects you, and the discriminant is the one number that decides which of three forms the solution has.
Fit the constants last, from two numbers. A second-order equation has two arbitrary constants, and the starting value alone leaves one of them free. Keep y(0) and y′(0) apart: the second is the rate of the unknown at the start, and not the derivative of the first number.
Say which figures are exact. Roots, constants and residuals are exact fractions or surds. A value of sin 1, cos 1 or e^(−2) is irrational, and is printed rounded, with ≈, and said to be rounded.
What it does not cover
Equations with a forcing term. This course solves the homogeneous equation a·y″ + b·y′ + c·y = 0; a right-hand side that is not zero belongs to Oscillators, Damping and Resonance. Variation of parameters, the general tool when a guess does not work, is named here and not taught.
Variable coefficients. If a, b or c depends on t, as in the Euler–Cauchy equation t²·y″ + b·t·y′ + c·y = 0, then e^(rt) is no longer a solution and the method does not apply. Series solutions and special functions such as Bessel's are a different subject and are not here.
Proofs of existence and uniqueness. The course uses the fact that two initial values fix exactly one solution, and the Wronskian lesson shows why for the cases it builds; it does not prove the general theorem.
Irrational constants. The lab fits the two constants only when the roots are rational or the complex pair has rational parts, and it refuses the rest with the reason. That is a limit of the arithmetic, and the method itself works for any roots.
Equations of third order and higher. The same substitution gives a cubic or worse characteristic equation, and the Subject stays with second order and with two-by-two systems, where the trace and determinant are complete.