# Solve: dy/dx+ y = xy^3 .??

This is non-linear but we can make it linear using a Bernoulli substitution. Here we let:

Integrating both sides, using IBP on RHS:

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This is a first-order nonlinear ordinary differential equation. To solve it, we can use the method of separation of variables.

- Rewrite the equation in the form ( \frac{dy}{dx} = xy^3 - y ).
- Factor out (y) on the right side to get ( \frac{dy}{dx} = y(x y^2 - 1) ).
- Now, separate variables: ( \frac{1}{y} dy = (x y^2 - 1) dx ).
- Integrate both sides. The integral of ( \frac{1}{y} dy ) is ( \ln|y| ), and the integral of ( (x y^2 - 1) dx ) requires a substitution.
- Solve the resulting integrals to find the general solution.

This process will lead to the solution of the given differential equation.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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