solve dy/dx-y=e^x/x given that y(e)=0

Question

please help me to answer this step by step., differential equation,

Answer 1

Let y=pq where p and q are functions of x.

dy/dx=pq'+qp' and dy/dx=pq+e^x/x replacing y in the given DE by pq.

Equating these two expressions for dy/dx, q=q' (dq/dx=q; dq/q=dx; ln(q)=x; q=e^x)

qp'=e^xp'=e^x/x; dp/dx=1/x; dp=dx/x; p=ln(x).

y=pq=ke^xln(x) (k=constant of integration); but y=0 when x=e, so 0=ke^e and k=0.

y=e^xln(x)

CHECK: dy/dx=e^x/x+e^xln(x)=e^x/x+y; dy/dx-y=e^x/x