Questions: Separate the variables and hence solve the differential equation [ y(x+1)=(x^2+2 x) fracd yd x ]

Solution Steps

To solve the given differential equation \( y(x+1) = (x^2 + 2x) \frac{dy}{dx} \), we need to separate the variables and then integrate both sides.

1. Rewrite the equation to separate the variables \( y \) and \( x \).
2. Integrate both sides with respect to their respective variables.
3. Solve for \( y \) as a function of \( x \).

We start with the differential equation given by:

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

Rearranging the equation allows us to separate the variables \( y \) and \( x \):

\[ \frac{dy}{y} = \frac{x^2 + 2x}{x + 1} dx \]

Integrating both sides gives us:

\[ \int \frac{dy}{y} = \int \frac{x^2 + 2x}{x + 1} dx \]

The left side integrates to:

\[ \ln |y| + C_1 \]

For the right side, we can simplify the integrand:

\[ \frac{x^2 + 2x}{x + 1} = x + 1 \]

Thus, the integral becomes:

\[ \int (x + 1) dx = \frac{x^2}{2} + x + C_2 \]

Combining the results from both integrals, we have:

\[ \ln |y| = \frac{x^2}{2} + x + C \]

Exponentiating both sides yields:

\[ y = e^{C} e^{\frac{x^2}{2} + x} \]

Letting \( C_1 = e^{C} \), we can express the solution as:

\[ y = C_1 e^{\frac{x^2}{2} + x} \]

Final Answer