Isolate derivative y': Given the differential equation 2xy' - ln(x^2) = 0, we want to find the general solution y(x). Let's start by isolating the derivative y'. 2xy' = ln(x^2) y' = ln(x^2) / (2x)Simplify ln(x^2): Now, we notice that ln(x^2) can be simplified using logarithm properties. ln(x^2) = 2ln(x) So, we can rewrite the equation as: y' = 2ln(x) / (2x)Cancel 2's: Simplifying the equation by canceling the 2's, we get: y' = ln(x) / x Now we have a separable differential equation.Integrate both sides: To solve the separable differential equation, we integrate both sides with respect to x. ∫y' dx = ∫(ln(x) / x) dxIntegrate known integral: The left side of the equation integrates to y. The right side is a known integral. y = ∫(ln(x) / x) dx The integral of ln(x) / x with respect to x is ln(x)^2 / 2 + C, where C is the constant of integration. y = ln(x)^2 / 2 + CFind general solution: We have found the general solution of the differential equation. The general solution is y(x) = ln(x)^2 / 2 + C.

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2 x y^{\prime}-\ln x^{2}=0

Isolate derivative $y'$ : Given the differential equation $2xy' - \ln(x^2) = 0$ , we want to find the general solution $y(x)$ . Let's start by isolating the derivative $y'$ .
$2xy' = \ln(x^2)$
$y' = \frac{\ln(x^2)}{2x}$
Simplify $\ln(x^2)$ : Now, we notice that $\ln(x^2)$ can be simplified using logarithm properties. $\ln(x^2) = 2\ln(x)$ So, we can rewrite the equation as: $y' = \frac{2\ln(x)}{2x}$
Cancel $2$ 's: Simplifying the equation by canceling the $2$ 's, we get: $\newline$ $y' = \frac{\ln(x)}{x}$ $\newline$ Now we have a separable differential equation.
Integrate both sides: To solve the separable differential equation, we integrate both sides with respect to $x$ . $\newline$ $\int y' \, dx = \int (\frac{\ln(x)}{x}) \, dx$
Integrate known integral: The left side of the equation integrates to $y$ . The right side is a known integral. $y = \int(\ln(x) / x) dx$ The integral of $\ln(x) / x$ with respect to $x$ is $\ln(x)^2 / 2 + C$ , where $C$ is the constant of integration. $y = \ln(x)^2 / 2 + C$
Find general solution: We have found the general solution of the differential equation. $\newline$ The general solution is $y(x) = \frac{\ln(x)^2}{2} + C$ .