Question

\( \left[\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right] \frac{d x}{d y}=1 \)

Answer 1

After taking dy/dx on other side it can be written as

dy/dx + y/√x = e^-2√x

This is an common liner differential equation of form

dy/dx + Py = Q

Here P is 1/√x and Q is e^-2√x

We solve this equation by this formula

ye^∫Pdx = ∫(Qe^∫pdx​ _{)dx  +c}

Therefore

ye^∫dx/√x = ∫(e^-2√x )(e^∫dx/√x)

ye^2√x = x​​​​​​^{​​​​​}​​​​