We are given the equation:

$\frac{1}{2y} \ln |y^2| = x^2 - x + c$

Step 1: Simplify $\ln |y^2|$

Since $\ln |y^2| = \ln y^2 = 2 \ln |y|$, we rewrite the equation as:

$\frac{1}{2y} \cdot 2 \ln |y| = x^2 - x + c$

$\frac{\ln |y|}{y} = x^2 - x + c$

Step 2: Introduce Integration

This equation is difficult to solve directly. We use integration to solve for $y$:

$\int \frac{\ln |y|}{y} dy = \int (x^2 - x + c) dx$

Using substitution, let $u = \ln |y|$, so that $du = \frac{1}{y} dy$. Then:

$\int du = \int (x^2 - x + c) dx$

$\ln |y| = \frac{x^3}{3} - \frac{x^2}{2} + cx + C_1$

where $C_1$ is the integration constant.

Step 3: Exponentiate Both Sides

Taking the exponential on both sides:

$|y| = e^{\frac{x^3}{3} - \frac{x^2}{2} + cx + C_1}$

Since $e^{C_1}$ is just another constant, say $C_2$, we rewrite:

$y = C_2 e^{\frac{x^3}{3} - \frac{x^2}{2} + cx}$

Thus, the final solution in explicit $y$-form is:

$y = C e^{\frac{x^3}{3} - \frac{x^2}{2} + cx}$

where $C = C_2$ is an arbitrary constant.

---

Would you like further clarification on any step?

Related Questions:

1. How do you solve differential equations using separation of variables?
2. What are the properties of logarithmic functions?
3. How does integration by substitution work?
4. What is the significance of the integration constant in solving equations?
5. How do exponential functions behave under differentiation and integration?

Tip:

When solving for $y$ explicitly, always check if exponentiation or logarithmic properties can simplify the equation!