Question

Determine the particular solution of x dy = y In y dx, given x= 2 when y = e.​

Answer 1

Answer:

I don't know the answer what is the answer sorry

Answer 2

xdy=y(lny)dx

∫dx/x=∫dy/(ylny)

Let ln(y)=t

So dt/dy=1/y

dt=dy/y, now replacing this in above equation:

∫dx/x=∫dt/t

ln(x)=ln(t)+c

ln(x)=ln(lny)+c

Now its given that when y=e, x=2, putting these values in the equation:

ln(2)=ln(ln(e))+c

ln(2)=ln(1)+c

c=ln(2) , (as ln(1)=0)

ln(x)=ln(lny)+ln(2)... [Using property: ln(a)+ln(b)=ln(a.b)]

ln(x)=ln[2lny]

x=2ln(y)

or x=ln(y²)... [Using property: a(lnb)=ln(bᵃ)