https://file.aiquickdraw.com/mathbot/file/1726665176542znonxnas.jpg

The equation provided is:

\[
x \, dy - y \, dx = x^2 \, dy
\]

We need to solve this first-order differential equation. Let's rewrite it and simplify.

### Step 1: Rearrange the equation

\[
x \, dy - y \, dx = x^2 \, dy
\]

Move all terms involving \(dy\) to one side and those involving \(dx\) to the other:

\[
x \, dy - x^2 \, dy = y \, dx
\]

Factor out \(dy\) on the left-hand side:

\[
(x - x^2) \, dy = y \, dx
\]

### Step 2: Separate variables

Now we divide both sides by \(y \, (x - x^2)\) to separate the variables:

\[
\frac{dy}{y} = \frac{dx}{x - x^2}
\]

### Step 3: Simplify and integrate both sides

We can now integrate both sides. Let me perform the integration:

\[
\int \frac{dy}{y} = \int \frac{dx}{x(1 - x)}
\]

The left side is straightforward:

\[
\ln |y| = \int \frac{dx}{x(1 - x)}
\]

We will use partial fraction decomposition for the right-hand side.

\[
\frac{1}{x(1 - x)} = \frac{A}{x} + \frac{B}{1 - x}
\]

Let me solve for \(A\) and \(B\).The partial fraction decomposition of \( \frac{1}{x(1 - x)} \) gives:

\[
\frac{1}{x(1 - x)} = \frac{1}{x} + \frac{1}{1 - x}
\]

### Step 4: Integrate both sides

Now we can rewrite the integral as:

\[
\int \frac{1}{x} \, dx + \int \frac{1}{1 - x} \, dx
\]

Both integrals are standard:

\[
\ln |x| - \ln |1 - x| = \ln \left( \frac{|x|}{|1 - x|} \right)
\]

Thus, the equation becomes:

\[
\ln |y| = \ln \left( \frac{|x|}{|1 - x|} \right) + C
\]

### Step 5: Solve for \(y\)

Exponentiate both sides to eliminate the logarithms:

\[
y = C' \frac{x}{1 - x}
\]

where \(C' = e^C\) is a constant. Thus, the general solution is:

\[
y = \frac{C x}{1 - x}
\]

This is the solution to the differential equation.

Would you like further details or explanations on any step? Here are some related questions to explore:

1. How can we confirm that this solution satisfies the original differential equation?
2. What method can be used to find a particular solution for given initial conditions?
3. Can this method be applied to other types of first-order linear equations?
4. How does partial fraction decomposition help in solving differential equations?
5. What are other techniques for solving first-order differential equations?

**Tip:** Always check if the solution you found satisfies the boundary conditions or initial values if provided.

Solve the differential equation x dy - y dx = x^2 dy using first-order differential equation methods.

Learn how to solve the first-order differential equation x dy - y dx = x^2 dy using separation of variables and partial fraction decomposition. This step-by-step solution is ideal for students studying undergraduate mathematics and 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