To solve the differential equation, let's carefully go through each step and find the error in your method.
Starting from the given differential equation:
dy/dx - y/x = 1/x + y + 1
Step 1 is correct. We rearrange the terms to separate the variables:
dy/dx = (y + 1 + xy + x)/x
Now, let's find the error in step 2. You have multiplied both sides by "x" and obtained:
x dy = (y + 1 + xy + x)dx
This step is incorrect. Instead of multiplying both sides by "x," we need to multiply both sides by dx to keep the separation of variables intact.
Correcting the error in step 2, we have:
dy = ((y + 1)dx + xy dx)/x
Now, let's simplify the equation using common denominators:
dy = (ydx + dx + xydx)/x
Next, factor out the dx term:
dy = (y + 1 + xy)dx/x
Now, multiply both sides by x:
x dy = (y + 1 + xy) dx
So, the corrected equation after step 2 is:
x dy = (y + 1 + xy) dx
Now, let's move on to step 3, where we integrate both sides with respect to x:
∫x dy = ∫(y + 1 + xy) dx
The integral on the left side is simply y:
xy + C1 = ∫(y + 1 + xy) dx
Integrating the right side:
xy + C1 = ∫y dx + ∫1 dx + ∫xy dx
The integral of y dx gives yx, the integral of 1 dx gives x, and the integral of xy dx gives (1/2)x^2y:
xy + C1 = yx + x + (1/2)x^2y + C2
Combining the constants into a single constant, let's rewrite the equation as:
xy - yx + (1/2)x^2y = x + C
Now, let's rearrange terms to isolate y:
xy - yx + (1/2)x^2y - (1/2)x^2y = x + C
yx = x + C
y = (x + C) / x
So, the correct solution to the differential equation is:
y = (x + C) / x
where C is the constant of integration.