Question

The particular solution of the differential equation \(e \frac{dy}{dx} = (x + 1)\), \(y(0) = 3\), is:

• A. \(y = x\log(x) - x + 2\)
• B. \(y = (x + 1)\log(x + 1) - x + 3\)
• C. \(y = (x + 1)\log(x + 1) + x - 3\)
• D. \(y = x\log(x) + x - 2\)

Hint:

When solving differential equations, verify the solution by substituting back into the original equation.

Answer 1

The Correct Option is B

The differential equation \(e \, dy = (x + 1) \, dx\) requires variable separation and integration. Integrating both sides yields \[\int e \, dy = \int (x + 1) \, dx \quad \Rightarrow \quad y = (x + 1)\log(x + 1) - x + C.\]Using the initial condition \(y(0) = 3\), we find the constant of integration: \[3 = (0 + 1)\log(0 + 1) - 0 + C \quad \Rightarrow \quad C = 3.\]The particular solution is therefore: \[y = (x + 1)\log(x + 1) - x + 3.\]