← Differential Equations

Unit 1: What a Differential Equation Is

What it means for a function to be the unknown, and how a direction field shows every solution before you solve anything.

8 topics · 31 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

1.1 What a Differential Equation Actually Is

  1. How it worksHow does a differential equation encode a rule for a rate of change instead of a single number?
  2. Worked exampleShow that y = 3e2x actually solves the equation y' = 2y.
  3. Common mistakeWhy isn't solving y' = 2y about isolating y the way you isolate x in an algebra equation?
  4. Going furtherWhat changes when the unknown rate depends on two variables instead of one?
Your turn1 of 2
A function y satisfies y' = 5y. If y(0) = 7, find y'(0).

1.1 What It Means to Solve a Differential Equation

  1. How it worksHow does one differential equation describe infinitely many different functions at once?
  2. How it worksHow does plugging a candidate function back into the equation prove it's actually a solution?
  3. Going furtherWhat happens to the family of solution curves for y' = ky as the constant k changes sign?
Your turn1 of 2
The general solution of y' = 2y is y = Ce2x. If C = 6, find y(3).

1.1 General Solutions Versus Particular Solutions

  1. How it worksHow does an initial condition pick out exactly one curve from the general solution's whole family?
  2. Worked exampleFind the particular solution of y' = 3x2 that passes through the point (1, 5).
  3. Common mistakeWhy isn't the general solution of a differential equation just one formula with no choices left in it?
  4. Going furtherWhat happens to the number of particular solutions when two initial conditions are imposed on a single first-order equation?
Your turn1 of 2
Find the particular solution y(x) of y' = 5x2 that satisfies y(2) = 9.

1.2 Verifying That a Function Solves a Differential Equation

  1. How it worksHow do you decide how many times to differentiate a candidate solution before plugging it back into a second-order equation?
  2. Worked exampleShow that y = x2 + Ce−x solves the equation y' + y = x2 + 2x for any constant C.
  3. Worked exampleShow whether y = sin(x) solves the equation y'' + y = 0.
  4. Going furtherWhat happens to the check when a proposed function solves the equation at some points but not others?
Your turn1 of 2
For y = 5sin(2x), find the value of y''(x) + 9y(x) at x = 2.

1.3 Classifying an Equation by Order and Linearity

  1. How it worksHow do you tell whether a differential equation is linear just by looking at how y and its derivatives appear?
  2. Worked exampleFind the order of y'' + 3xy' - y3 = sin(x) and whether it's linear or nonlinear.
  3. Common mistakeWhy isn't an equation nonlinear just because it has a nonlinear function like sin(x) sitting inside it?
Your turn1 of 2
Classify the equation y''' + 4x y'' - 4y = cos(x): what is its order, and is it linear or nonlinear?

1.4 Direction Fields

  1. How it worksHow does plotting a short line segment with slope f(x,y) at every grid point build a direction field?
  2. How it worksHow do solution curves thread through a direction field by always running tangent to the local segment?
  3. Going furtherWhat happens to a direction field's pattern when y' = f(x,y) is changed to y' = f(y) alone?
  4. Ties togetherHow does dragging a solution curve's starting point through a direction field connect to choosing an initial condition?
Your turn1 of 2
For the differential equation y' = 3x + 3y, find the slope of the direction field segment at the point (4, 2).

1.4 Sketching a Solution Curve from a Direction Field

  1. How it worksHow do you trace a solution curve through a direction field by following the segments like a trail of arrows?
  2. Worked exampleSketch the solution curve through the point (0, 1) in the direction field for y' = x - y.
  3. Common mistakeWhy isn't a direction field's arrow at a point just a suggestion the curve is free to ignore?
  4. Going furtherWhat happens to a sketched solution curve when it's extended backward instead of forward through the direction field?
  5. Ties togetherHow does sketching a curve through a direction field by hand connect to what a numerical method like Euler's method does automatically?
Your turn1 of 2
In the direction field for y' = x - 3y, find the slope of the solution curve exactly at its starting point (3, 4).

1.5 Initial Value Problems

  1. How it worksHow does an initial value problem combine a differential equation with a starting point to select one curve?
  2. Worked exampleSolve the initial value problem y' = 2y, y(0) = 5.
  3. Going furtherWhat changes about the solution to y' = 2y when the initial condition moves from y(0) = 5 to y(0) = -5?
  4. Common mistakeWhy doesn't specifying y(0) alone pin down a solution to a second-order differential equation?
Your turn1 of 2
Solve the initial value problem y' = 3y, y(0) = 6. Find y(t).