The given differential equation is:

$\frac{dy}{dx} = \frac{y^3}{x+1}.$

This is a separable differential equation, which we can solve by separating the variables $y$ and $x$. Let's solve it step by step.

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Step 1: Rearrange to separate variables

Rewriting the equation: $\frac{1}{y^3} \, dy = \frac{1}{x+1} \, dx.$

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Step 2: Integrate both sides

We integrate each side: $\int \frac{1}{y^3} \, dy = \int \frac{1}{x+1} \, dx.$

Left-hand side:

$\int \frac{1}{y^3} \, dy = \int y^{-3} \, dy = \frac{y^{-2}}{-2} = -\frac{1}{2y^2}.$

Right-hand side:

$\int \frac{1}{x+1} \, dx = \ln|x+1|.$

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Step 3: Combine results

After integrating, we have: $-\frac{1}{2y^2} = \ln|x+1| + C,$ where $C$ is the constant of integration.

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Step 4: Solve for $y^2$ (optional)

Multiply through by $-2$ to simplify: $\frac{1}{y^2} = -2\ln|x+1| - 2C.$

Let $C_1 = -2C$ (absorbing constants): $\frac{1}{y^2} = -2\ln|x+1| + C_1.$

Invert to find $y^2$: $y^2 = \frac{1}{-2\ln|x+1| + C_1}.$

Finally, take the square root: $y = \pm \sqrt{\frac{1}{-2\ln|x+1| + C_1}}.$

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This is the general solution. Let me know if you'd like further details or clarification!

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Related Questions:

1. How do we interpret constants of integration in solutions?
2. Can this equation be solved for specific initial conditions?
3. What happens if $x = -1$? Does the solution still hold?
4. How do we solve non-separable differential equations?
5. Can we visualize this solution graphically?

Tip:

When solving separable equations, always check if the variables are separable before attempting other methods!