kirill_kirill

To solve this differential equation, we'll use an integrating factor. First, we'll rearrange the equation to get y' on its own: xy' + y = x - 1 y' = (x - 1 - y)/x Now we'll find the integrating factor, which is e^(integral of x/x dx). The integral of 1/x is ln|x|, so the integrating factor is e^(ln|x|) = |x|. Multiplying both sides of the equation by the integrating factor gives: |x|y' + |x|y = |x|(x - 1) Now we can use the product rule to simplify the left-hand side: d/dx (|x|y) = |x|y' + y So the equation becomes: d/dx (|x|y) = |x|(x - 1) Integrating both sides with respect to x gives: |x|y = (1/2)x^2 - x + C where C is the constant of integration. Solving for y gives: y = (1/