Find the general solution of the differential equation y' = (y^2 + x^2) / 2x^2

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.

Discover the general solution to the differential equation y' = (y^2 + x^2) / 2x^2 using separation of variables and integration techniques. This problem involves key concepts in differential equations and requires knowledge of integration methods. Suitable for college-level students and enthusiasts of calculus.

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