Question

Solve the differential equation \( 2(y + 3) - xy \frac{dy}{dx} = 0 \); given \( y(1) = -2 \).

Hint:

For separable differential equations, rearrange to isolate \( x \) and \( y \) terms before integrating.

Answer 1

Step 1: Rewrite the equation in standard form. \[ 2(y + 3) - xy \frac{dy}{dx} = 0. \] Rearrange: \[ \frac{dy}{dx} = \frac{2(y + 3)}{xy}. \] Step 2: Separate the variables. \[ \frac{dy}{y + 3} = \frac{2dx}{x}. \] Step 3: Integrate both sides. \[ \int \frac{dy}{y + 3} = \int \frac{2dx}{x}. \] \[ \log |y + 3| = 2 \log |x| + C. \] Step 4: Solve for \( y \). \[ y + 3 = e^C x^2. \] Let \( e^C = C_1 \), so: \[ y = C_1 x^2 - 3. \] Step 5: Apply initial condition \( y(1) = -2 \). \[ -2 = C_1(1)^2 - 3. \] \[ C_1 = 1. \] Final Solution: \[ y = x^2 - 3. \]