Questions: Solve the initial value problem. x^2 dy/dx = (2x^2 - x - 4)/(x+1)(y+1), y(1)=3 Begin by separating the variables. Choose the correct answer below A. (y+1) dy = (2x^2 - x - 4)/x^2(x+1) dx B. dy/dx = (2x^2 - x - 4)/x^2(x+1)(y+1) C. x^2(x+1)/(2x^2 - x - 4) dy 1/(y+1) dx D. The equation is already separated. The solution is y^2/2 + y = ln x + ln x+1 + . (Type an implicit solution. Type an equation using x and y as the variables.)

Solution Steps

To solve the initial value problem, we start by separating the variables. This involves rearranging the given differential equation so that all terms involving \( y \) are on one side and all terms involving \( x \) are on the other side. Once separated, we integrate both sides to find the implicit solution. Finally, we use the initial condition \( y(1) = 3 \) to solve for the constant of integration.

To solve the differential equation \( x^2 \frac{dy}{dx} = \frac{2x^2 - x - 4}{(x+1)(y+1)} \), we first separate the variables. This involves rearranging the equation to isolate terms involving \( y \) on one side and terms involving \( x \) on the other side:

\[ (y+1) \, dy = \frac{2x^2 - x - 4}{x^2(x+1)} \, dx \]

This matches option A from the multiple-choice answers.

Next, we integrate both sides of the separated equation:

\[ \int (y+1) \, dy = \int \frac{2x^2 - x - 4}{x^2(x+1)} \, dx \]

The left side integrates to:

\[ \frac{y^2}{2} + y \]

The right side requires partial fraction decomposition and integration, which results in:

\[ \ln |x| + \ln |x+1| + C \]

We use the initial condition \( y(1) = 3 \) to solve for the constant \( C \). Substituting into the integrated equation:

\[ \frac{3^2}{2} + 3 = \ln |1| + \ln |1+1| + C \]

\[ \frac{9}{2} + 3 = 0 + \ln 2 + C \]

\[ \frac{15}{2} = \ln 2 + C \]

Solving for \( C \), we find:

\[ C = \frac{15}{2} - \ln 2 \]

Final Answer