# What is the general solution of the differential equation # (x^2+y^2)dx+(x^2-xy)dy = 0 #?

# y/x-2ln(y/x+1) = lnx + C #

We have:

We can rearrange this Differential Equation as follows:

So Let us try a substitution, Let:

Then:

This is now a separable DIfferential Equation, and so "separating the variables" gives us:

This is now a trivial integration problem, thus:

And restoring the substitution we get:

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See below.

Making the substitution

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To solve the differential equation ((x^2+y^2)dx + (x^2 - xy)dy = 0), we can start by rearranging it into a more manageable form. The goal is to find a solution (y(x)) or to express the relationship between (x) and (y) implicitly.

Given: [ (x^2 + y^2)dx + (x^2 - xy)dy = 0 ]

Rearrange it to separate variables or find an integrating factor that simplifies the equation. This equation is not immediately separable, nor is it exact as is. However, we can attempt to make it exact by identifying an integrating factor or through manipulation.

In this case, it's helpful to notice that both terms involve (x^2), suggesting we might want to look for symmetry or a way to combine terms efficiently. However, directly solving this as presented without a clear method to separate variables or an obvious integrating factor suggests we look at alternative methods such as substitution or transformation.

Given the symmetry in the equation and the mix of (x^2) and (y^2) terms, one might be tempted to use polar coordinates ((x = r\cos(\theta)), (y = r\sin(\theta))), but the mixed (xy) term complicates this approach directly.

Another approach could be to try and identify a function or form that this equation represents directly, but without clear separability, exactness, or a linear form, we fall back on intuition or advanced methods that could involve looking at the equation from the perspective of differential forms or leveraging more complex transformations.

To solve this equation directly here without stepping into methods that require a deeper dive into advanced mathematical theories or transformations that aren't immediately apparent from the equation as given, we would need more context or to apply a specific technique tailored to the equation's structure. In standard undergraduate courses, an equation of this form might lead to a teaching moment about recognizing when and how to apply certain techniques like finding an integrating factor for exact equations, using substitutions effectively, or recognizing patterns that suggest a move to polar coordinates or another coordinate system might simplify the problem.

Without applying a specific, advanced method or a clever substitution that isn't immediately apparent from the form of the equation given, we acknowledge this problem's complexity and the potential need for a deeper exploration of mathematical techniques beyond the basic separation of variables, integration factors, or simple transformations.

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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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- What is the general solution of the differential equation #xy'' = y'+x(y')^2#?

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