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How to solve the following ODE:

$xy'-y=\left(x^{2}+2x\right)\sin\left(3x\right)$.

That is: how we find a relation between $x$ and $y$ without any derivative or integral symbol in that relation.

If we use the standard formula, it might be difficult by calculating the antiderivative.

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    calculate $(y/x)'$ – 2017-01-21
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    The solution contains the [Sine Integral](http://mathworld.wolfram.com/SineIntegral.html), a non-elementary function. – 2017-01-21
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    How about the solution in some relation between $x$ and $y$? It might rule out Sine integral??? – 2017-01-21

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We have $$y'-\frac {1}{x} y =(x+2)\sin 3x $$ This is a linear differential equation. The integrating factor is $e^{\int -\frac {1}{x} dx} = e^{-\log x} =\frac {1}{x} $.

The solution is thus $$y (\frac {1}{x}) =\int \frac {1}{x}(x+2)\sin 3x dx$$ Can you take it from here?

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    How can we move forward from here? – 2017-01-21
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    As in the comments you get a Sine integral (Si). Anyways integrating by parts gives us $$(x+2)\operatorname {Si}(3x) -\int \operatorname {Si}(3x) dx$$ Now use the fact $\int \operatorname {Si}(x) dx = x\operatorname {Si}(x) +\cos x $. – 2017-01-21