The problem asks us to solve a first-order linear differential equation using Bernoulli's method. The image does not contain the actual differential equation, only the description. Therefore, I will provide a general solution strategy for solving a first-order differential equation with Bernoulli's method. We consider the general form: $\frac{dy}{dx} + P(x)y = Q(x)y^n$ where $n$ is a real number and $n \neq 0, 1$.

1. Problem Description

The problem asks us to solve a first-order linear differential equation using Bernoulli's method. The image does not contain the actual differential equation, only the description. Therefore, I will provide a general solution strategy for solving a first-order differential equation with Bernoulli's method. We consider the general form:

$\frac{dy}{dx} + P(x)y = Q(x)y^n$

where $n$ is a real number and $n \neq 0, 1$.

2. Solution Steps

Step 1: Divide the equation by $y^n$:

$y^{-n} \frac{dy}{dx} + P(x)y^{1-n} = Q(x)$

Step 2: Make the substitution $z = y^{1-n}$.

Then $\frac{dz}{dx} = (1-n) y^{-n} \frac{dy}{dx}$.

Therefore, $y^{-n} \frac{dy}{dx} = \frac{1}{1-n} \frac{dz}{dx}$.

Step 3: Substitute these into the equation:

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

Multiply by $(1-n)$:

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

This is a first-order linear differential equation in $z$.

Step 4: Find the integrating factor $\mu(x)$:

$\mu(x) = e^{\int (1-n)P(x) dx}$

Step 5: Multiply the equation by $\mu(x)$:

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

Step 6: Observe that the left side is the derivative of $\mu(x)z$:

$\frac{d}{dx}(\mu(x)z) = \mu(x)(1-n)Q(x)$

Step 7: Integrate both sides with respect to $x$:

$\int \frac{d}{dx}(\mu(x)z) dx = \int \mu(x)(1-n)Q(x) dx$

$\mu(x)z = \int \mu(x)(1-n)Q(x) dx + C$

Step 8: Solve for $z$:

$z = \frac{1}{\mu(x)} \int \mu(x)(1-n)Q(x) dx + \frac{C}{\mu(x)}$

Step 9: Substitute back $z = y^{1-n}$:

$y^{1-n} = \frac{1}{\mu(x)} \int \mu(x)(1-n)Q(x) dx + \frac{C}{\mu(x)}$

Finally, solve for $y$:

$y = \left( \frac{1}{\mu(x)} \int \mu(x)(1-n)Q(x) dx + \frac{C}{\mu(x)} \right)^{\frac{1}{1-n}}$

3. Final Answer

The general solution to the Bernoulli differential equation $\frac{dy}{dx} + P(x)y = Q(x)y^n$ is:

$y = \left( \frac{1}{\mu(x)} \int \mu(x)(1-n)Q(x) dx + \frac{C}{\mu(x)} \right)^{\frac{1}{1-n}}$, where $\mu(x) = e^{\int (1-n)P(x) dx}$.