Question

If $X=x+h, Y=y+k$ transforms $\frac{dy}{dx} = \frac{2x+3y-7}{3x+2y-8}$ to a homogeneous differential equation, then $(h,k)=$

• A. $(1,2)$
• B. $(2,1)$
• C. $(7,8)$
• D. $(8,7)$

Hint:

Homogenizing Rational Expressions:

• Translate $x \to x + h$, $y \to y + k$ to eliminate constants.
• Solve the resulting linear system.

Answer 1

The Correct Option is B

To make RHS homogeneous, eliminate constants: \[ 2h + 3k = 7, \quad 3h + 2k = 8 \] Solving: \[ 6h + 9k = 21, \quad 6h + 4k = 16 \Rightarrow 5k = 5 \Rightarrow k = 1 \Rightarrow h = 2 \] So, $(h,k) = (2,1)$.