Question

If the general solution of the differential equation \[ \frac{dy}{dx} + \frac{2x+2y-1}{x+y-5} = 0 \] is \[ ax+by-9\log(x+y+p)=c, \] and \(b,a,p\) are in G.P., then the common ratio of this G.P. is

• A. \(3\)
• B. \(2\)
• C. \(\dfrac12\)
• D. \(\dfrac13\)

Hint:

For equations involving \[ x+y, \] use the substitution \[ \boxed{z=x+y}, \] so that \[ \boxed{\frac{dz}{dx}=1+\frac{dy}{dx}}, \] which usually converts the equation into a separable form.

Answer 1

The Correct Option is B

Step 1: Rewrite the differential equation. Given, \[ \frac{dy}{dx} = -\frac{2x+2y-1}{x+y-5}. \] Let \[ z=x+y. \] Then, \[ \frac{dz}{dx} = 1+\frac{dy}{dx} = 1-\frac{2z-1}{z-5} = -\frac{z+4}{z-5}. \] Hence, \[ \frac{z-5}{z+4}\,dz = -dx. \]

Step 2: Integrate. Now, \[ \frac{z-5}{z+4} = 1-\frac9{z+4}. \] Therefore, \[ \int\left(1-\frac9{z+4}\right)dz = -\int dx. \] This gives \[ z-9\log(z+4) = -x+C. \] Substituting \[ z=x+y, \] we obtain \[ 2x+y-9\log(x+y+4)=C. \] Comparing with \[ ax+by-9\log(x+y+p)=c, \] we get \[ a=2,\qquad b=1,\qquad p=4. \]

Step 3: Find the common ratio. The numbers \[ b,\;a,\;p \] are \[ 1,\;2,\;4, \] which form a G.P. with common ratio \[ \frac21 = \frac42 = 2. \] Hence, \[ \boxed{2}. \] Thus, \[ \boxed{(B)} \] is the correct answer.