Differential Equations: Models, Methods, and Population Dynamics Free Online FlashCards

What is a differential equation?

A differential equation is an equation involving an unknown function and one or more of its derivatives.

What does an initial condition determine?

An initial condition selects one particular solution from a family of solutions, creating an initial-value problem.

Translate: rate proportional to current population

“Rate proportional to current size” translates to dP/dt = kP, the exponential growth or decay model.

When is a first-order equation separable?

A separable equation can be written as dy/dx = f(x)g(y), allowing the variables to be rearranged onto opposite sides before integration.

Why check equilibrium solutions before separation?

Check separately for constant equilibrium solutions before dividing by a factor involving y; division can remove solutions where that factor is zero.

What does a slope field show for dy/dx = f(x,y)?

At (x,y), the segment’s slope is f(x,y). A solution curve passing through that point is tangent to the segment.

State Euler’s method update formulas

Euler’s method uses xₙ₊₁ = xₙ + h and yₙ₊₁ = yₙ + h f(xₙ,yₙ).

Use Euler’s method for y′=x+y, y(0)=1, h=0.1

Starting with x₀=0, y₀=1 and h=0.1 gives y₁=1.1, y₂=1.22, and y₃=1.362; thus y(0.3)≈1.362.

What is the exponential growth or decay solution?

For dP/dt = kP, the solution is P(t)=P₀eᵏᵗ, where P₀=P(0).

What is the exponential doubling-time formula?

The doubling time is T_d = ln(2)/k for exponential growth.

State the logistic differential equation

The logistic model is dP/dt = rP(1−P/K), where r is the intrinsic growth rate and K is the carrying capacity.

What are the logistic equilibrium populations?

The logistic equilibria are P=0 and P=K, because dP/dt equals zero at both values.