https://r2.erweima.ai/mathbot/image/1727102650938mnv9vpfg.jpg

The provided image shows a differential equation:

\[
\frac{dy}{dx} = \frac{y}{x}
\]

This is a **separable differential equation**, meaning that variables \(y\) and \(x\) can be separated on different sides of the equation. Let's solve it step by step.

### Step 1: Separate the variables
We want to rewrite the equation to separate \(y\) and \(x\) on different sides of the equation:
\[
\frac{dy}{y} = \frac{dx}{x}
\]

### Step 2: Integrate both sides
Integrate both sides of the equation:
\[
\int \frac{1}{y} dy = \int \frac{1}{x} dx
\]

This gives:
\[
\ln |y| = \ln |x| + C
\]
where \(C\) is the constant of integration.

### Step 3: Solve for \(y\)
Exponentiate both sides to remove the logarithms:
\[
|y| = e^{\ln |x| + C}
\]
This simplifies to:
\[
|y| = e^C \cdot |x|
\]
Let \(e^C\) be a new constant, say \(A\), so the solution becomes:
\[
|y| = A|x|
\]

### Step 4: Remove the absolute value
The general solution is:
\[
y = A x
\]
where \(A\) is a constant.

Thus, the solution to the differential equation is \( y = A x \), which represents a family of straight lines passing through the origin, where \(A\) is a constant.

Would you like further details or have any questions? Here are five related questions to deepen your understanding:

1. How do we determine the constant \(A\) if initial conditions are given?
2. Can this method be used to solve non-separable differential equations?
3. How does the solution of this differential equation change if boundary conditions are provided?
4. What is the geometric interpretation of the solution \(y = Ax\)?
5. How do separable and non-separable differential equations differ in terms of solution techniques?

**Tip:** When solving separable differential equations, always check if you can rewrite the equation by separating variables before integrating.

Assinale a alternativa que contenha o resultado da equação diferencial: \( \frac{dy}{dx} = \frac{y}{x} \)

Learn how to solve the separable differential equation dy/dx = y/x with a step-by-step solution. This problem demonstrates key concepts in differential equations and integration. Suitable for undergraduate students.

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