To find the general solution of the differential equation $y' = \frac{y^2 + x^2}{2x^2}$, we can use the method of separation of variables.

The given differential equation is: $y' = \frac{y^2 + x^2}{2x^2}$

First, rewrite the equation in a separable form: $\frac{dy}{dx} = \frac{y^2 + x^2}{2x^2}$

Simplify the right-hand side: $\frac{dy}{dx} = \frac{y^2}{2x^2} + \frac{x^2}{2x^2} = \frac{y^2}{2x^2} + \frac{1}{2}$

Now, separate the variables $y$ and $x$: $\frac{dy}{y^2 + \frac{1}{2}y^2} = \frac{dx}{x}$

Combine the $y$ terms: $\frac{dy}{\frac{y^2}{2} + \frac{x^2}{2}} = \frac{dx}{x}$ $\frac{dy}{y^2 + x^2} = \frac{dx}{x}$

Next, integrate both sides: $\int \frac{2 dy}{y^2 + 1} = \int \frac{dx}{x}$

Use the substitution $u = y / x$, then $y = ux$ and $dy = u dx$.

The equation becomes: $\frac{du}{u^2 + 1} = \frac{1}{x} dx$

Integrate both sides: $\int \frac{du}{u^2 + 1} = \int \frac{dx}{x}$

We know that: $\int \frac{du}{u^2 + 1} = \arctan(u)$ $\int \frac{dx}{x} = \ln|x| + C$

Thus: $\arctan\left(\frac{y}{x}\right) = \ln|x| + C$

Exponentiating both sides to solve for $y$: $y = x \tan(\ln|x| + C)$

So, the general solution to the differential equation $y' = \frac{y^2 + x^2}{2x^2}$ is:

$y = x \tan(\ln|x| + C)$

Would you like any details or have any questions? Here are some related questions you might want to explore:

1. How do you solve a first-order separable differential equation?
2. What are the common methods for solving first-order differential equations?
3. How do you find a particular solution for a given initial condition?
4. What is the significance of integrating factors in solving differential equations?
5. How do you solve differential equations using substitution methods?

Tip: Always verify the solution by differentiating it and substituting back into the original differential equation to ensure it satisfies the equation.