Solve the following differential equations

$\displaystyle x \frac{dy}{dx}+2y=\frac{2}{x} \ln(x)$. Differential equations is one of my toughest subjects, could someone please help with solving

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This is linear. Are you aware of the general method to solve it(with the integrating factor)? – Ishan Banerjee Mar 6 '13 at 12:11
This one is linear. There's a method for solving these which you surely have access to. – Gerry Myerson Mar 6 '13 at 12:11
First step I'm suppose to divide everything by x, then? – user58315 Mar 6 '13 at 12:13

1. Divide by $x$. The eq. becomes: $$\dfrac{dy}{dx} + 2 \dfrac{y}{x} = \dfrac{2}{x^2} \ln{x}$$
2. Calculate the Integrating Factor. $$\text{I.F.} = e^{\int{\dfrac{2}{x} dx}} = e^{2\ln{x}} = x^2$$
3. Multiply the whole eq. by $x^2$. $$x^2 \dfrac{dy}{dx} + 2xy = 2 \ln{x}$$
4. Let $u = x^2y$, so $$\dfrac{du}{dx} = x^2 \dfrac{dy}{dx} + 2xy$$
5. The original eq. now becomes: $$\dfrac{du}{dx} = 2 \ln{x}$$
6. Integrate both sides, w.r.t $x$: $$u = 2 \int{ \ln{x} \text{ } dx} = 2 \left( x \ln{x} - x\right)$$
7. Replace $u$ in terms of $x$ and $y$. $$y = \dfrac{2}{x} \cdot \left( \ln{x} - 1 \right) + \text{I}$$

where, $I$ is the integrating constant.

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We get $$y'+y\frac{2}{x}=\frac{2\ln x}{x^2}$$

which is a first order linear ordinary differential equation. This is of the form $$y' + y\cdot P(x)=Q(x)$$ whose solution is given by $$y\cdot e^{\int P(x)dx}=\int \left[Q(x)\cdot e^{\int P(x)dx} dx\right] + C.$$

So, the solution of the given DE is $$y\cdot e^{\int \frac{2}{x}dx}=\int \left[\frac{2\ln x}{x^2}\cdot e^{\int \frac{2} {x}dx} dx\right] + C.$$ Hope you can finish the solution on your own.:)

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Got it, thanks! – user58315 Mar 6 '13 at 12:43
@user58315, Welcome.:) – juniven Mar 6 '13 at 12:46
@user58315, You have not yet accepted my answer.:) – juniven Mar 6 '13 at 12:48

If we do not know any systematic methods, we can rewrite the equation as $x^2\frac{dy}{dx}+2xy=2\ln x$ and recognize the left-hand side as the derivative of $x^2y$. It follows that $$x^2y=\int 2\ln x\,dx=2x\ln x-2x+C.$$

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