The given equation is $y'' = \frac{y'}{x}$. Let's solve it step-by-step.

Step 1: Substitute $y' = v$, where $v = \frac{dy}{dx}$

This substitution changes the second-order equation into a first-order equation in terms of $v$: $y'' = \frac{d}{dx}(y') = \frac{dv}{dx} = \frac{dv}{dx}$ Thus, the equation becomes: $\frac{dv}{dx} = \frac{v}{x}$

Step 2: Solve the first-order differential equation for $v$

Rewriting: $\frac{dv}{v} = \frac{dx}{x}$ Integrating both sides: $\ln |v| = \ln |x| + C_1$ where $C_1$ is the constant of integration. Simplify: $v = C_2 x$ where $C_2 = e^{C_1}$ is a positive constant.

So, $y' = v = C_2 x$.

Step 3: Solve for $y$

Now integrate $y' = C_2 x$: $y = \int C_2 x \, dx = \frac{C_2 x^2}{2} + C_3$ where $C_3$ is another constant of integration.

Final Answer:

The general solution of the differential equation is: $y = \frac{C_2 x^2}{2} + C_3$ or equivalently: $y = C_1 x^2 + C_2,$ where $C_1 = \frac{C_2}{2}$ and $C_2 = C_3$.

Would you like more details on any specific step?

Related Questions:

1. What happens if the constants $C_1$ or $C_2$ are set to specific values?
2. How can this solution be verified?
3. What are practical examples where such differential equations arise?
4. How would the solution change if an additional term $+f(x)$ was added to the right-hand side?
5. Can we solve this equation using numerical methods for a specific case?

Tip:

Always substitute your solution back into the original equation to verify its correctness!