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Solve the system of differential equations $$\frac{dx}{dt}+\frac{-4x+2y}t=-9$$ $$\frac{dy}{dt}+\frac{x-5y}t=3$$ in $t\ge 1$ subject to $x=0$ and $y=0$ at $t=1$.

It gives the hint [powers of $t$].

I have tried rearranging, differentiating and from the hint I thought to try series solution but nothing works.

Thank you for any help

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    It's from a past paper which could ask anything, I guess you could narrow down what to try by looking at the other questions and assuming they won't ask two questions on the same topic, but essentially it could be any part. – 2017-01-11
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    you can eliminate one variable from the system – 2017-01-11
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    @Dr.SonnhardGraubner thanks so far, but how? – 2017-01-11
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    then you can solve one differential equation. – 2017-01-11
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    what equation do you get? – 2017-01-11
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    have you got the solution? – 2017-01-11
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    @moo How did you get $6^{th}$ order equations in $t$? – 2017-01-11
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    @Dr.SonnhardGraubner No, I'm not sure how to eliminate a variable – 2017-01-11

1 Answers 1

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Considering the following linear combinations:

$$\frac{dx}{dt}+\frac{dy}{dt}-\frac{3(x+y)}{t}+6=0$$

$$\frac{dx}{dt}-2\frac{dy}{dt}-\frac{6(x-2y)}{t}+15=0$$

Can you proceed?

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    Looking at it I think so. I haven't done it yet but I think a couple of substitutions here may work? If I have interpreted this correctly, the way I would proceed will not really use the hint, but maybe if I'm wrong. Thank you – 2017-01-11