The solution curve of the differential equation y (d x)/(d y)=x(log ₑ x-log ₑ y+1), x>0, y>0 passing through the point (e, 1) is

Solution

Maths sir · 4 steps

Everything depends on x/y, so put x = vy and the equation separates neatly.

1. Spot the form. Since logₑ x - logₑ y = logₑ x/y, the equation is dx/dy = x/y(logₑ x/y + 1). The right side is a function of x/y only, so it is homogeneous with y as the independent variable.
2. Substitute x = vy. Then dx/dy = v + ydv/dy, which gives v + ydv/dy = v logₑ v + v. The v cancels on both sides, leaving ydv/dy = v logₑ v.

   ∫ dv/(v logₑ v) = ∫ dy/y ⇒ logₑ|logₑ v| = logₑ y + logₑ c

3. Remove the logs. This gives |logₑ v| = cy, that is, |logₑ x/y| = cy. At (e, 1) we get v = e, so |logₑ e| = 1 = c × 1, which means c = 1.
4. So. The solution curve is |logₑ x/y| = y, which is option (c).

The idea to keep. When the right side depends only on x/y, substitute x = vy and separate. A log constant becomes a multiplying constant, so expect a form like |log v| = cy.

Answer from the NTA final answer key. Solution by PrepWiser.