Solve the initial-value problem: $x(x+1)\frac{dy}{dx}+xy=1$, $y\left(e\right)=1$. What is the solution $y\left(x\right)$?

• A. $y\left(x\right)=\frac{1}{x}+\frac{e^{-x}}{x}$
• B. $y\left(x\right)=\frac{1}{x}+\frac{e^{-x}}{x(x+1)}$
• C. $y\left(x\right)=\frac{1}{x}+\frac{e^{-x}}{x+1}$
• D. $y\left(x\right)=\frac{1}{x}+\frac{e^{-x}}{x(x+1)}$

Answer: (B)

Verified step by step guidance

1. Step 1: Rewrite the given differential equation x(x + 1) * (dy/dx) + x * y = 1 in a standard form. Divide through by x(x + 1) to isolate dy/dx. This gives (dy/dx) + (y / (x + 1)) = 1 / (x(x + 1)).
2. Step 2: Recognize that this is a first-order linear differential equation. The standard form is (dy/dx) + P(x)y = Q(x), where P(x) = 1 / (x + 1) and Q(x) = 1 / (x(x + 1)).
3. Step 3: Find the integrating factor, μ(x), which is e^(∫P(x)dx). Compute ∫P(x)dx = ∫(1 / (x + 1))dx = ln|x + 1|. Thus, μ(x) = e^(ln|x + 1|) = x + 1.
4. Step 4: Multiply through the differential equation by the integrating factor μ(x) = x + 1. This transforms the equation into (x + 1)(dy/dx) + y = (x + 1) * (1 / (x(x + 1))). The left-hand side becomes d/dx(y * μ(x)), or d/dx(y * (x + 1)).
5. Step 5: Integrate both sides with respect to x. The left-hand side integrates to y * (x + 1), and the right-hand side integrates ∫(1 / x)dx = ln|x|. Solve for y(x) using the initial condition y(e) = 1 to determine the constant of integration.