Use an appropriate substitution to solve the equation y'-(9/x)y=y^5/(x^18)
and find the solution that satisfies y(1)=1

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. 

Use an appropriate substitution to solve the equation y'-(9/x)y=y^5/(x^18) and find the solution that satisfies y(1)=1

Step-by-step solution to the Bernoulli differential equation y' - (9/x)y = y^5/(x^18), using substitution to linearize the equation and solve it with the initial condition y(1) = 1. Ideal for college students studying advanced calculus or differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation