# What is the general solution of the differential equation #dy/dx+x^4y=3x^4 (x>0)#? Express the answer in explicit form.

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I understand that the equation is of the form that uses the integrating factor method:

#dy/dx+g(x)y=h(x)# ,

with #g(x)=x^4# , and #h(x)=3x^4# .

The integrating factor

#p(x)=exp(intx^4dx)=exp((x^5)/5)=e^((x^5)/5)#

The general solution is

#y=1/(p(x))(intp(x)h(x) dx)#

#=1/e^((x^5)/5)(int3x^4e^((x^5)/5)dx)#

..... and going on from here is where I get confused... very basic step by step explanation from here on would be very gratefully received.

I understand that the equation is of the form that uses the integrating factor method:

with

The integrating factor

The general solution is

#=1/e^((x^5)/5)(int3x^4e^((x^5)/5)dx)#

..... and going on from here is where I get confused... very basic step by step explanation from here on would be very gratefully received.

The solution is

I see a simpler solution.

We separate the variables

Integrating both sides

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The answer is

We can apply your method

Integrating both sides

Therefore,

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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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