Question

The particular solution of the differential equation \[ \frac{dx}{dy} = \frac{\sin y(1 + y \cot y)}{x \log(x^2 e)}, \quad y(1) = 0 \] Options:

• A. \( y \sin y = x^2 \log x \)
• B. \( y^2 \sin y = \log x \)
• C. \( y = \frac{e^2}{\sin e} (x - 1) \)
• D. \( y = e^2 \sec x \)

Hint:

For solving differential equations with separation of variables, ensure that you properly handle the logarithmic terms and integrate both sides carefully. Apply initial conditions to obtain the particular solution.

Answer 1

The Correct Option is A

We are given the differential equation: \[ \frac{dx}{dy} = \frac{\sin y(1 + y \cot y)}{x \log(x^2 e)} \] We also know the initial condition \( y(1) = 0 \).
Step 1: Simplify the equation.
First, simplify the expression for \( \log(x^2 e) \): \[ \log(x^2 e) = \log(x^2) + \log(e) = 2 \log x + 1 \] Substitute this into the differential equation: \[ \frac{dx}{dy} = \frac{\sin y (1 + y \cot y)}{x (2 \log x + 1)} \] Step 2: Solve the differential equation.
We use the method of separation of variables. First, multiply both sides by \( x (2 \log x + 1) \) to separate the variables: \[ x (2 \log x + 1) \, dx = \sin y (1 + y \cot y) \, dy \] Now, integrate both sides: \[ \int x (2 \log x + 1) \, dx = \int \sin y (1 + y \cot y) \, dy \] The integration on the left-hand side leads to: \[ \int x (2 \log x + 1) \, dx = x^2 \log x \] For the right-hand side, solving the integral gives: \[ \int \sin y (1 + y \cot y) \, dy = y \sin y \] Step 3: Apply the initial condition.
Using the initial condition \( y(1) = 0 \), substitute into the equation: \[ y \sin y = x^2 \log x \] This is the required particular solution. Final Answer: \[ \boxed{y \sin y = x^2 \log x} \]