Question

An appropriate substitution to solve the differential equation $\frac{dx}{dy}=\frac{x^2\log \left({\displaystyle \frac{x}{y}}\right)-x^2}{xy\log \left({\displaystyle \frac{x}{y}}\right)}$ is ________________.

Answer 1

Given: $\frac{dx}{dy}=\frac{x^2\log \left({\displaystyle \frac{x}{y}}\right)-x^2}{xy\log \left({\displaystyle \frac{x}{y}}\right)}$

To solve this differential equation, we use the method of homogeneous differential equation.
Thus we put, x = vy.
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Hence, an appropriate substitution to solve the differential equation $\frac{dx}{dy}=\frac{x^2\log \left({\displaystyle \frac{x}{y}}\right)-x^2}{xy\log \left({\displaystyle \frac{x}{y}}\right)}$ is $\underline{\frac{x}{y}=v}$.