How do you solve the differential #y'xln(x)=y#?

Answer 1

#y=Ce^( 1/4x^2(2 ln x - 1) #, where #C >0#, and so is y.

To make ln x real, x > 0.

Separating variables and integrating,

#int 1/y dy=int x ln x dx#. So,
#ln y #
#= int ln x d(x^2/2)#
#=1/2x^2 ln x-1/2 int x^2 d(ln x)#
#=1/2x^2 ln x- 1/2 intx^2/x dx#
#=1/2x^2 ln x- 1/2 intx^2/x dx#
#=1/2x^2 ln x -x^2/4+A#

So,

#y =e^Ae^( 1/4x^2(2 ln x - 1) #
#=Ce^( 1/4x^2(2 ln x - 1) #, where #C > 0#. and so is y.
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Answer 2

To solve the differential equation ( y'x\ln(x) = y ), you can use the method of separating variables. Rearrange the equation to isolate the variables ( y ) and ( x ), and then integrate both sides. The solution is:

[ y = Cx ]

where ( C ) is the constant of integration.

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Answer 3

To solve the differential equation ( y'x \ln(x) = y ), you can use separation of variables method. First, rewrite the equation in the form ( \frac{dy}{dx} = \frac{y}{x \ln(x)} ). Then, separate the variables ( y ) and ( x \ln(x) ) and integrate both sides with respect to their respective variables. This will involve integrating ( \frac{1}{y} , dy ) on one side and integrating ( \frac{1}{x \ln(x)} , dx ) on the other side. After integrating, solve for ( y ) to find the general solution of the differential equation.

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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.

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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