Final answer:
To solve the given initial-value problem x(x + 1) * dy/dx + xy = 1, y(e) = 1, we can use the method of integrating factors. The general solution is xy(x+1) = (1/3)x^3 + (1/2)x^2 + C, where C is the constant of integration.
Step-by-step explanation:
To solve the given initial-value problem: x(x + 1) dy/dx + xy = 1, y(e) = 1, we can use the method of integrating factors. First, we rearrange the equation to isolate dy/dx: dy/dx = (1 - xy) / (x(x+1)). Next, we identify the integrating factor, which is e^(∫(x+1)/x dx). Simplifying the integrating factor gives us e^(ln|x(x+1)|) = x(x+1). We then multiply the entire equation by the integrating factor, giving us x(x+1) * dy/dx + xy(x+1) = x(x+1). The left side of the equation is a product rule, which simplifies to d(xy(x+1))/dx = x(x+1). Now, we integrate both sides of the equation to find the general solution: ∫ d(xy(x+1))/dx dx = ∫ (x(x+1)) dx. This gives us xy(x+1) = (1/3)x^3 + (1/2)x^2 + C, where C is the constant of integration.