(X+y) dx + xdy =0
(x+y)dx = -x dy
-x dy /(x+y) dx = 1
-x / [ x(1 +y/x)] dx = 1
- dy /(1 +y/x) = dx
Let u =y/x then xu = y so x du/dx + u = dy/dx.
- (x du/dx + u) / ( 1+ u ) = 1
x du/dx + u = -(1 + u)
x du/dx + u = -1 - u
x du/dx + 2u = -1
(x/dx) du + 2u = -1
x/dx + (2u)/du = -1/du
x/dx = (-1 - 2u)/du
du/(-1-2u) = dx/x
(-1/2)log|-1 - 2u| = log|x| + C
Multiplying by -2:
log|-1-2u| = -2log|x| + C
log|-1-2u| = log|1/x^2| + C
The given equation is (x+y)dx + xdy = 0. This is a first-order linear differential equation. To solve it, we can use the method of integrating factors.
Step 1: Re-arrange the equation in the standard form: dy/dx + (x+y)/x = 0.
Step 2: Identify the coefficient of dy/dx, which is 1.
Step 3: Determine the integrating factor, denoted by μ, by multiplying the entire equation by the reciprocal of the coefficient of dy/dx. In this case, the integrating factor is μ = 1/x.
Step 4: Multiply both sides of the equation by μ = 1/x. This gives us (1/x) * dy/dx + (x+y)/(x^2) = 0.
Step 5: Simplify the equation to get dy/dx + (1/x)y + (1/x^2)x = 0.
Step 6: Notice that the left side of the equation can be written as d/dx (y/x) = 0. This implies that y/x is a constant.
Step 7: Solve y/x = C for y to get y = Cx, where C is a constant.
So, the general solution to the differential equation is y = Cx, where C is any constant.