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a) Find the general solution to the differential equation: $$2\frac{d^{2}y}{dx^{2}}-4\frac{dy}{dx}+20y=0.$$ b) Find the general solution to the differential equation: $$x^{2}\frac{d^{2}y}{dx^{2}}+4x\frac{dy}{dx}+2y=\frac{1}{x}.$$

I am having trouble with these two differential equations in a past paper I am going through. Thanks in advance for any replies!

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    Part (a) at least is an absolutely standard 'just follow what's in the notes' question. What do your lecture notes say about such equations? Where do you get stuck when you follow the method you have learned?2012-08-29
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    For part (b), you can make this into a constant-coefficient equation with the change of independent variable $x = e^t$.2012-08-29
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    Alternatively, you can use Reduction of Order or Variation of Parameters after solving the homogeneous equation (which is an Euler differential equation).2012-08-29

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