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Calculus II (Integration & Series) Unit 6: Differential equations: Euler & logistic

Euler's method approximates solutions step by step. The logistic model describes growth that levels off at a carrying capacity.

Covers: Euler's method, logistic growth

Study guide

Many differential equations can't be solved exactly, so we approximate. Euler's method steps along the slope field: starting from (x₀, y₀), use the slope at the current point to step forward by h: yₙ₊₁ = yₙ + h·f(xₙ, yₙ), and xₙ₊₁ = xₙ + h. Repeat to reach the target x. Each step follows a tangent line, so error builds up; smaller step sizes give better approximations. If solution curves are concave up, Euler's method underestimates, and if concave down it overestimates.

Logistic growth models populations limited by resources. The logistic differential equation is dP/dt = kP(1 − P/L), where k is a growth constant and L is the carrying capacity. When P is small, growth is nearly exponential; as P approaches L, growth slows to zero.

Key facts can be read directly from the equation. The equilibrium solutions are P = 0 and P = L. If 0 L, it decreases toward L. Either way, lim t→∞ P(t) = L. The population grows fastest when P = L/2, which is the inflection point of the logistic curve.

The solution has the form P(t) = L/(1 + Ae^(−kt)), where A depends on the initial population. This S-shaped curve appears in populations, the spread of rumors and diseases, and product adoption.

Many models, including the logistic one, are solved by separation of variables. Move every P-term to the side with dP and every t-term to the side with dt, integrate (using partial fractions for the logistic case), add C, and apply the initial condition. To get d²y/dx² from a differential equation, differentiate dy/dx with respect to x, treat y as a function of x, and replace each dy/dx with its formula. The sign of the result tells you the concavity, and so whether an Euler or tangent-line estimate is an over- or underestimate.

The logistic equation can also appear as dP/dt = kP(L − P) or dP/dt = aP − bP². In every form, the carrying capacity is the nonzero P-value that makes the right side zero (L, or a/b), and growth is fastest at half that value. Equilibrium solutions are constant values where dy/dt = 0. They appear as horizontal segments in a slope field, and other solutions can't cross them.

Key ideas

• Euler's method: follow the tangent line for a small step, then repeat.
• Smaller steps give better Euler approximations.
• Logistic growth: fastest when P = L/2, and P approaches L.
• The logistic solution is an S-shaped curve.

Formulas and rules

Common mistakes

• Using the slope at the new point instead of the current point in Euler's method.
• Thinking logistic growth is fastest at the start.

Words to know

Euler's method update

yₙ₊₁ = yₙ + h·f(xₙ, yₙ)

Logistic differential equation

dP/dt = kP(1 − P/L)

Logistic inflection point

Growth is fastest at P = L/2.

Carrying capacity (L)

The limit of P as t → ∞ in logistic growth.

Euler's method step size

Smaller h gives a more accurate approximation.

Euler's method

yₙ₊₁ = yₙ + h·f(xₙ, yₙ): step along tangent lines.

Step size (h)

The x-increment per Euler step.

Logistic solution

P(t) = L/(1 + Ae^(−kt)), an S-shaped curve.

Logistic equilibria

P = 0 (unstable) and P = L (stable).

Sample practice questions

25 more questions like these, from level 1 to 5, are in the practice set.