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What is the general solution of the differential equation # dy/dx = (x+y)/x #?

Answer 1

Evelyn Agard

# y = xln|x| + Cx #

We have:

# dy/dx = (x+y)/x # ..... [A]

If we use the suggested substitution, #y=vx# then differentiating wrt #x# and applying the product rule we have:

# dy/dx = (v)(d/dx x) + (d/dx v)(x) # # \ \ \ \ \ = (v)(1) + (dv)/dx x # # \ \ \ \ \ = v + x(dv)/dx #

Substituting this result into the initial differential equation [A] we get:

# v + x(dv)/dx = (x+vx)/x # # :. v + x(dv)/dx = 1+v # # :. x(dv)/dx = 1 # # :. (dv)/dx = 1/x #

Which has reduced the equation to a trivial First Order separable equation, which we can "separate the variables" to get:

# int \ dv = int \ 1/x \ dx #

And if we integrate we get:

# v = ln|x| + C #

Restoring the earlier substitution, we get:

# y/x = ln|x| + C #

Leading to the General Solution:

# y = xln|x| + Cx #

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Answer 2

Emily Ammerman

The general solution of the given differential equation dy/dx = (x+y)/x is y = cx + x, where c is an arbitrary constant.

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Answer from HIX Tutor

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.