Solve each differential equation using the given boundary...

Solve each differential equation using the given boundary condition. (Hint: Rewrite each differential equation in the form $g(y) d y=f(x) d x$ and integrate both sides.
$$
\frac{d y}{d x}=\frac{x^{2}+x+1}{x}, y=0 \text { when } x=1
$$

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Key Concepts

Initial Value Problems

An initial value problem specifies not only the differential equation but also an initial condition that the solution must satisfy. By applying this initial condition after obtaining the general solution, one can determine the specific constant of integration, thereby finding the particular solution that meets the given criteria.

Integration

Integration is used to determine the antiderivatives of the separated functions. By integrating both sides of the equation, one develops a general solution that encapsulates the family of all possible solutions to the differential equation.

Separation of Variables

This method involves manipulating a differential equation so that one side contains only the dependent variable (and its differential) while the other side contains only the independent variable (and its differential). This separation allows us to integrate each side with respect to its own variable, which is a standard approach for solving many first order differential equations.

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