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The technique of separation of variables is a powerful method used to solve differential equations. When you have an equation like \( \frac{d y}{d x}=y(1+y) \), separation of variables allows you to rearrange it so each variable appears on opposite sides of the equation. This way, you can independently integrate both sides.

In this case, the original differential equation is separated by dividing both sides by \( y(1+y) \) and multiplying by \( dx \). The result is \( \frac{1}{y(1+y)} \, dy = dx \). This creates an equation that can be integrated with respect to its own variable, making it possible to solve the differential equation step by step.

Separation of variables is particularly useful because it simplifies the integration process by reducing the problem to two independent integrals. This method can tackle a variety of differential equations, provided they can be rewritten with the separated forms.

When faced with complex rational expressions during integration, partial fraction decomposition can be a lifesaver. This technique involves breaking down a complicated fraction into simpler fractions that are easier to integrate. In the context of this problem, the expression \( \frac{1}{y(1+y)} \) is decomposed into \( \frac{1}{y} - \frac{1}{1+y} \).

To perform partial fraction decomposition, you consider the denominator's factors. Here, \( y(1+y) \) is expressed as two separate terms, \( y \) and \( 1+y \). By writing the initial fraction as a sum or difference of individual fractions corresponding to these terms, the resulting expressions become more straightforward to handle during integration.

This method is particularly useful for rational expressions where the degree of the numerator is less than the degree of the denominator. It’s a staple in calculus because it simplifies integrals, making them more approachable for students.

Integration is a vital part of solving differential equations, turning derivative expressions into function statements. Here, once variables in the differential equation are separated and partial fraction decomposition is applied, integration can proceed smoothly.

For the decomposed expressions \( \frac{1}{y} \) and \( \frac{1}{1+y} \), the integration results in natural logarithms: \( \int \frac{1}{y} \, dy = \ln |y| \) and \( \int \frac{1}{1+y} \, dy = \ln |1+y| \).

These results are obtained using a standard rule of integration for \( \frac{1}{u} \), which equals \( \ln |u| + C \). This technique is common when dealing with rational expressions.

Additionally, integrating \( dx \) results in \( x + C \), where \( C \) is the constant of integration. This constant captures the family of solutions related to initial conditions or boundary values.

Mastering integration techniques is crucial in calculus, as they enable the connection between derivatives and antiderivatives, opening doors to solving a wide array of mathematical problems.