if `y+x(dy)/(dx)=x(phi(xy))/(phi'(xy))` then `phi(xy)` is equation to

Options

A. `ke^(x^(2)//2)`
B. `ke^(y^(2)//2)`
C. `ke^(xy//2)`
D. `ke^(xy)`

Answer

A

Explanation

Put `xy=v`, i.e., `y+x(dy)/(dx)=x(phi(v))/(phi^(')(v))`
`therefore (phi^(')(v))/(phi(v))dv=xdx`
Integrating, we get
`logphi(v) = x^(2)/2 + logk`
or `logphi(v)/k = x^(2)/2`
or `phi(v) = ke^(x^(2)//2)`
or `phi(xy)=ke^(x^(2)//2)`