Final answer:
To solve the given differential equation by separation of variables, we can rewrite it in a simplified form and integrate both sides. Then, solve for y using logarithmic properties and exponentiation. The solution to the differential equation is y = e^(x + C - ln|x - 9|) + 1.
Step-by-step explanation:
To solve the given differential equation by separation of variables, we can rewrite it as:
dy/(xy + 4y - x) = (x - 9)²/(x - 9)² - 4/(xy - 9y + x) dx
Now we can integrate both sides. Integrate the left side with respect to y and the right side with respect to x:
∫1/(x + 4) dy = ∫(x - 9)²/(x - 9)² - 4/(xy - 9y + x) dx
The integral on the left side is ln|x + 4| + C1, and the integral on the right side can be simplified to: ∫(x - 9)²/(x - 9)² - 4/(x - 9)(y - 1) dx
We can further simplify the right side as: ∫1 - 4/(x - 9)(y - 1) dx
This can be solved using partial fractions. The integral on the right side becomes: ∫1 dx - ∫(1/(x - 9)(y - 1)) dx
The first integral is simply x + C2. The second integral can be solved using a u-substitution. Let u = x - 9, so du = dx. The integral becomes: ∫(1/u(y - 1)) du = ln|u(y - 1)| + C3 = ln|x - 9||y - 1| + C3.
Now we can equate the left and right sides and solve for y:
ln|x + 4| + C1 = x + C2 + ln|x - 9||y - 1| + C3
This equation involves logarithmic terms, so we can simplify it using logarithmic properties. We can also combine the constants into a single constant C:
ln|x + 4| = x + ln|x - 9||y - 1| + C
Finally, we can exponentiate both sides to solve for y:
|x + 4|e^(x + C) = |x - 9||y - 1|
Dividing both sides by |x - 9| gives:
e^(x + C - ln|x - 9|) = |y - 1|
Since the absolute value of a positive number is always positive, we can simplify further:
e^(x + C - ln|x - 9|) = y - 1
So the solution to the differential equation is y = e^(x + C - ln|x - 9|) + 1.