To solve the differential equation:

$y' - \frac{9}{x} y = \frac{y^5}{x^{18}}$

with the initial condition $y(1) = 1$, we will use an appropriate substitution and solve it step by step.

Step 1: Recognize the Structure of the Equation

This is a Bernoulli differential equation. The general form of a Bernoulli equation is:

$y' + P(x)y = Q(x)y^n$

where $n$ is a constant. Comparing the given equation with this form, we identify:

• $P(x) = -\frac{9}{x}$
• $Q(x) = \frac{1}{x^{18}}$
• $n = 5$

Step 2: Substitution to Linearize the Equation

In a Bernoulli equation, we make the substitution $z = y^{1-n} = y^{1-5} = y^{-4}$. Taking the derivative of $z$ with respect to $x$, we get:

$z = y^{-4} \quad \Rightarrow \quad \frac{dz}{dx} = -4y^{-5}y'$

Rewriting the equation in terms of $z$ and $y$:

• ( y' = -\frac{1}{4} z' y^5 = -\frac{1}{4pop model.