Unit 12: Differential Equations
Equations built from a rate of change, solved by finding the function hiding behind the rate.
6 topics · 26 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.
Verifying a Solution to a Differential Equation
- How it worksHow does substituting a candidate function and its derivatives back into the equation confirm or kill it?
- Worked exampleFind whether y = e2x is a solution to the differential equation y' - 2y = 0.
- Worked exampleFind whether y = x2 + C solves the differential equation y' = 2x for every constant C.
- Common mistakeWhy doesn't a differential equation have just one solution the way a normal equation has one answer?
Your turn1 of 2
Which function solves the differential equation y' - 3y = 0?Separable Differential Equations
- How it worksHow does separating variables and integrating both sides turn dy/dx = f(x)g(y) into an implicit solution?
- Worked exampleFind the general solution to the differential equation dy/dx = xy.
- Worked exampleFind the particular solution to dy/dx = 2xy with y(0) = 3.
- Common mistakeWhy doesn't every differential equation allow its x's and y's to be separated onto opposite sides?
Your turn1 of 2
Find the particular solution y(x) to dy/dx = 3xy with y(0) = 4.Direction Fields
- How it worksHow does drawing a short line segment with slope f(x, y) at each grid point build up a direction field?
- Worked exampleFind the slope indicated by the direction field for dy/dx = x - y at the point (2, 1).
- Common mistakeWhy doesn't a direction field show you exactly which solution curve passes through a given point?
- Going furtherWhat happens to a direction field near a point where dy/dx equals zero along an entire curve?
- Going furtherWhat happens to a direction field when the differential equation is autonomous, depending only on y?
Your turn1 of 2
Find the slope indicated by the direction field for dy/dx = x - 3y at the point (4, 5).Euler's Method
- How it worksHow does repeatedly nudging forward by step size h times the local slope build up Euler's method?
- Worked exampleFind the Euler's method approximation for y(0.2) given dy/dx = x + y, y(0) = 1, with step size 0.1.
- Common mistakeWhy doesn't Euler's method give the exact solution even though each step follows the true local slope?
- Going furtherWhat happens to Euler's method's accuracy as the step size h shrinks toward zero?
Your turn1 of 2
Starting at y(0) = 3, take two Euler's method steps of size 0.286 for dy/dx = x + 2y. What is the resulting approximation for y?Exponential Growth and Decay Models
- How it worksHow does solving dy/dt = ky produce the exponential model y = y0 ekt?
- Worked exampleFind how long it takes a substance with a half-life of 10 years to decay to 25 percent of its original amount.
- Worked exampleFind the growth constant k for a bacteria population that doubles every 3 hours.
- Common mistakeWhy doesn't exponential growth mean a quantity increases by the same amount in every equal time interval?
Your turn1 of 2
Find how long it takes a substance with a half-life of 15 years to decay to 42 percent of its original amount.The Logistic Growth Model
- How it worksHow does the logistic equation dy/dt = ky(1 - y/M) build in a carrying capacity that exponential growth ignores?
- Worked exampleFind the logistic model's solution given carrying capacity 1000, growth rate k = 0.3, and initial population 50.
- Worked exampleFind the equilibrium population sizes of the logistic equation dy/dt = 0.4y(1 - y/500).
- Common mistakeWhy doesn't a logistic population ever exceed its carrying capacity even when it starts by growing quickly?
- Ties togetherHow does the logistic equation reduce to plain exponential growth when the carrying capacity is treated as infinite?
Your turn1 of 2
A population follows the logistic model with carrying capacity 1196, growth rate k = 0.395, and initial population 95. Write the population P(t) as a function of time t.