Find the general solution of the given differential equation. x dy dx − y = x2 sin(x) give the largest interval over which the general solution is defined. (think about the implications of any singular points. enter your answer using interval notation.

The general solution of a differential equation is to write y as a function of x.

Given

Divide through by x

Let P be function of x. Such that:

So, we have:

Calculate the integrating factor I(x).

So, we have:

Substitute

Rewrite as:

Integrate

So, we have:

Introduce .

So, we have:

Multiply both sides by dx

Integrate with respect to x

Multiply through by x

So, the general solution is: , and the interval is

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Divide both sides by . In doing so, we force any possible solutions to exist on either or (the "positive" interval in such a situation is usually taken over the "negative" one) because cannot be 0 in order for us to do this.

Condense the left side as the derivative of a product, then integrate both sides and solve for :

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step-by-step explanation:

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