1
Write an equation in standard form and read p and q
Divide every term by the coefficient of y′, name p and q with their signs, say whether the equation is homogeneous, and say why t²·y is linear and y² is not.
Course · Differential Equations
First-Order Linear Equations
One equation does a great deal of work. A tank filling, a capacitor charging, a loan being repaid and a cooling cup of coffee are all y′ + a·y = b with different letters, and every one of them is a constant plus a gap that shrinks by a fixed factor in equal times. This course solves that equation three ways: as a shifted exponential, by multiplying through by an integrating factor that makes the left side a derivative, and as a homogeneous solution plus one particular solution found by a guess whose coefficients are matched exactly. The cases that are exact throughout are the ones with p constant or a/t, where the factor is an exponential or a power of t. The last lesson turns to the method rather than the equation: Euler's factor 1 − a·h is an exact fraction, and it leaves the interval from −1 to 1 at a step size you can compute.
y′ + p(t)·y = q(t) p and q read off first
y′ + a·y = b: steady state y* = b/a
μ = e^(∫p dt), (μ·y)′ = μ·q
y = y_p + y_h fit C last
undetermined coefficients: exact matching
Euler on y′ = −a·y: factor 1 − a·h
stable only while h < 2/a
What this course is about
- Subject
- Differential Equations
- Lessons
- 8
- Format
- Interactive
Overview
What you will be able to do
By the end you can put a first-order linear equation in standard form, solve it by whichever of the three methods suits it, fit the constant from a starting value, check every answer by substitution, and say at what step size Euler's method stops following a decay.
2
Solve y′ + a·y = b and name the steady state
Find b/a, write the starting gap, put it on e^(−at), say whether the steady state attracts, and compare Euler's exact steps with the closed form.
3
Build and use an integrating factor
Construct μ with μ′ = p·μ, show that the left side becomes (μ·y)′, and solve a case with μ = tᵃ entirely in fractions.
4
Split a solution into homogeneous and particular parts
Find one particular solution by a guess with matched coefficients, add the homogeneous solution, and fit the constant after the sum.
5
Fit particular solutions to polynomial, exponential and sinusoidal forcing
Choose the form from the forcing, match coefficients exactly, and handle the resonant case where the forcing matches the homogeneous solution.
6
Bound the step size at which Euler's method fails
Compute the factor 1 − a·h, derive h < 2/a for decay, and show that the backward factor decays at every step size.
Lessons
8 lessons
The form first, because every method assumes it, then the constant-coefficient case, which needs nothing more than a shifted exponential. The integrating factor and the split into homogeneous and particular parts are two solutions of one equation, and the forcing lessons make the second of them a method. Applications come next, and the course ends with the step size, where the exact arithmetic of the labs exposes what a decimal would hide.
The Standard FormDivide by the coefficient of y′, read off p and q, and the equation is in the one shape every method on this course assumes.Not marked
Constant Coefficients and the Steady StateSolve y′ + a·y = b as a shifted exponential, name the steady state b/a, and set Euler's exact steps beside the closed form.Not marked
The Integrating FactorConstruct μ = e^(∫p dt), show that (μ·y)′ = μ·q is the product rule read backwards, and solve a case with μ = tᵃ entirely in fractions.Not marked
Homogeneous Plus ParticularSplit a solution into the homogeneous part and one particular part, explain why adding them is allowed, and fit the constant last.Not marked
Polynomial Forcing and Undetermined CoefficientsGuess a polynomial of the forcing's degree with every lower term, match coefficients from the top down, and solve the exact system.Not marked
Exponential and Sinusoidal ForcingFit A·e^(bt) and A·cos(ωt) + B·sin(ωt) particular solutions with exact coefficients, and multiply by t when the forcing matches the homogeneous solution.Not marked
Circuits, Tanks and LoansWrite three situations as one equation y′ + a·y = b, read the steady state in each, and compute when a loan balance reaches zero.Not marked
Stiffness: When the Step Is Too BigShow Euler's factor 1 − a·h on y′ = −a·y, derive the bound h <; 2/a for decay, and show that the backward factor decays at every step size.Not marked
Recommended background
The previous course, Equilibria, Stability and Phase Lines, for the idea that a constant solution can attract or repel, together with Separable Equations, Growth and Decay for y′ = k·y and its exponential solution, Differential Equations and Euler's Method for Euler's steps as exact fractions, the product rule from Rates of Change and the Derivative, and antiderivatives of powers from Accumulation and the Integral.
Practice suggestions
Read the standard form before doing anything else. Every formula on this course is written for y′ with coefficient 1, and the commonest error is a formula applied to an equation that has not been divided yet. The lab states p and q in its status line so that you can compare them with your own.
Check every solution by substituting it into the equation, not by comparing it with a picture. The check is one derivative and one addition, it is exact, and it catches a wrong sign in the exponent or a constant divided by the wrong factor, which a plot of a decaying curve does not.
Fit the constant last. In every method here the general solution has the constant in it, and the starting value is a statement about the whole solution. Putting it in before the particular part is added is the error the course names more than once.
Say which figures are exact. The fractions in the Euler columns are exact arithmetic on an approximate method, and a value of e^(−2) is rounded and printed with ≈. The two sit side by side in the labs on purpose.
What it does not cover
Variation of parameters as a general tool. The integrating factor is the first-order version and is complete here; the second-order version is named in Second-Order Linear Equations as the method that works when a guess does not.
Equations whose coefficient p(t) is not a constant or a/t. The integrating factor e^(∫p dt) exists for any continuous p, and the lessons state it in general, but the labs solve only the two families whose factor is exact, and the course does not teach integrals that need a logarithm to print.
Nonlinear equations that can be turned linear by a substitution, such as Bernoulli and Riccati equations. They are techniques for finding closed forms that the exact checks of this Subject cannot verify.
Existence and uniqueness. A linear equation with continuous p and q has exactly one solution through each starting point on an interval where they are continuous; the course uses this and does not prove it.
Stiffness as a numerical-analysis topic. The last lesson shows one exact factor, its bound, and the backward method's answer to it. Stability regions, implicit Runge–Kutta methods and adaptive step sizes are not here.