# How do you find the particular solution to 2xy'-lnx^2=0 that satisfies y(1)=2?

Jan 25, 2017

Use the separation of variables method.
Integrate.
Use $y \left(1\right) = 2$ to solve for the value of the integration constant.

#### Explanation:

Separate variables:

$\mathrm{dy} = \ln \frac{{x}^{2}}{2 x} \mathrm{dx}$

Integrate:

$y = \int \ln \frac{{x}^{2}}{2 x} \mathrm{dx}$

$y = \int \ln \frac{x}{x} \mathrm{dx}$

$y = {\ln}^{2} \left(x\right) + C$

Use $y \left(1\right) = 2$ to solve for c:

$2 = {\ln}^{2} \left(1\right) + C$

$2 = C$

The particular solution is:

$y = {\ln}^{2} \left(x\right) + 2$