To solve the initial value problem $\frac{dy}{dx} = -\frac{y}{x}$ with the initial condition $y(1) = 1$, we proceed as follows:

Step 1: Rewrite the differential equation

The equation can be rewritten in separable form: $\frac{1}{y} \, dy = -\frac{1}{x} \, dx$

Step 2: Integrate both sides

Integrate both sides: $\int \frac{1}{y} \, dy = \int -\frac{1}{x} \, dx$

The left side integrates to $\ln|y|$ and the right side integrates to $-\ln|x| + C$, where $C$ is the constant of integration: $\ln|y| = -\ln|x| + C$

Step 3: Solve for $y$

Exponentiate both sides to solve for $y$: $|y| = e^C \cdot \frac{1}{|x|}$ Let $e^C = k$, a constant. Then: $y = \frac{k}{x}$

Step 4: Apply the initial condition $y(1) = 1$

Substitute $x = 1$ and $y = 1$ into the solution to find $k$: $1 = \frac{k}{1} \implies k = 1$

Step 5: Final solution

The solution to the initial value problem is: $y = \frac{1}{x}$

Summary

The solution is: $y = \frac{1}{x}$

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Here are five related questions you might find helpful:

1. What are separable differential equations and how are they solved?
2. How does the integration constant $C$ affect general solutions to differential equations?
3. What are other methods to solve first-order differential equations, such as substitution?
4. How do you handle initial conditions for differential equations with multiple variables?
5. Can you solve a similar differential equation, such as $\frac{dy}{dx} = y/x$?

Tip: Always verify your solution by substituting it back into the original differential equation.