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📘 Key Concepts: Bernoulli Equations

A Bernoulli equation has the form:

$$\frac{dy}{dx} + P(x)\,y = Q(x)\,y^n \quad (n \neq 0, 1)$$

Substitute $z = y^{1-n}$ to reduce it to a linear first-order equation in $z$:

$$\frac{dz}{dx} + (1-n)\,P(x)\,z = (1-n)\,Q(x)$$

Steps:

  1. Identify $n$ (the power of $y$ on the right).
  2. Let $z = y^{1-n}$; then $\dfrac{dz}{dx} = (1-n)\,y^{-n}\dfrac{dy}{dx}$.
  3. Divide the original equation by $y^n$ and substitute — it becomes linear in $z$.
  4. Solve using the integrating factor method; back-substitute $y = z^{1/(1-n)}$.

Problem: Bernoulli Equation ($n = 3$)

Solve $\dfrac{dy}{dx} + y = x\,y^3$.

DE – Bernoulli – Problem 1 – Setup DE – Bernoulli – Problem 1 – Diagram DE – Bernoulli – Problem 1 – Diagram
Solution Solution Solution Solution

Problem: Bernoulli with $x$ Coefficient ($n = 2$)

Solve $x\,\dfrac{dy}{dx} + y = x^3 y^2$.

DE – Bernoulli – Problem 2 – Setup DE – Bernoulli – Problem 2 – Diagram DE – Bernoulli – Problem 2 – Diagram
Solution Solution Solution Solution

Problem: Bernoulli with Exponential Right Side

Solve $\dfrac{dy}{dx} - y = y^2 e^x$.

DE – Bernoulli – Problem 3 – Setup DE – Bernoulli – Problem 3 – Diagram DE – Bernoulli – Problem 3 – Diagram
Solution Solution Solution Solution

Problem: Logistic Population Model

The logistic model for population growth is $\dfrac{dP}{dt} = kP\!\left(1 - \dfrac{P}{M}\right)$, where $M$ is the carrying capacity.
Given: $k = 0.3$, $M = 10{,}000$, and $P(0) = 1{,}000$.
a. Solve for $P(t)$ using the Bernoulli equation method.
b. Find the population at $t = 5$ years.
c. Find when the population reaches 5,000.

DE – Bernoulli – Problem 4 – Setup DE – Bernoulli – Problem 4 – Diagram DE – Bernoulli – Problem 4 – Diagram
Solution Solution Solution Solution

Problem: Bernoulli Equation with n = 2

Solve dy/dx + y = y2.

Let $v=y^{1-2}=y^{-1}$, so $v'=-y^{-2}y'$.

$$v'-v=-1$$
$$v=1+Ce^x$$

Answer: $y=\dfrac{1}{1+Ce^x}$.