Question

If \( u = x^2 + y^2 + 2xy + 2x + 2y \) and \( v = e^{x+y} \) are functionally dependent, then the relation between \( u \) and \( v \) is:

• A. \( u^2 = v^2 + \log uv \)
• B. \( u = (\log v)^2 + 2 \log v \)
• C. \( u = 2 \log v + 2 \log u \)
• D. \( v = (\log u)^2 + 2 \log u \)

Hint:

In functional dependence problems, if you see terms like \( x^2+2xy+y^2 \) alongside \( e^{x+y} \), the common link is almost always the linear term \( x+y \).

Answer 1

The Correct Option is B

Step 1: Test functional dependence with the Jacobian.
Two functions \( u(x,y) \) and \( v(x,y) \) are functionally dependent exactly when their Jacobian vanishes everywhere. Here \( u_x = 2x+2y+2 \) and \( u_y = 2y+2x+2 \) are equal, and \( v_x = v_y = e^{x+y} \), so the Jacobian determinant \( u_x v_y - u_y v_x \) is automatically zero, confirming that \( u \) can be written purely in terms of \( v \).

Step 2: Introduce a single variable for the dependence.
Since both functions only depend on \( x \) and \( y \) through the sum \( x+y \), set \( t = x+y \). Then \( v = e^{t} \) so \( t = \ln v \), and \( u = x^2+y^2+2xy+2x+2y = t^2+2t \).

Step 3: Substitute back.
Replacing \( t \) with \( \ln v \) gives \[ u = (\ln v)^2 + 2\ln v \] which matches option (B).