I was just looking over the method of variation of parameters again and remembered one part that made me always wonder if this method can be improved in any way. Here's a link for reference: http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx(adsbygoogle = window.adsbygoogle || []).push({});

Now everything stated seems to make sense, but I am wondering if this assumption is known to make this method ineffective at times or not:

## u_{1}^{'} y_{1} + u_{2} ^{'} y_{2} = 0 ##

If I'm not mistaken, this is simply done to simplify the equations later on. That is perfectly fine, but I'm just wondering, doesn't this restriction impose unnecessary constraints on the solutions? By doing this, aren't we only restricting ourselves to a few classes of solutions instead of all of them? In some cases, isn't it possible that:

## u_{1}^{'} y_{1} + u_{2} ^{'} y_{2} = C ## where ## C \neq 0 ##?

I guess I just never quite understood the validity of this assumption and am wondering if this causes any major consequences when using this method. Any explanations for why or why not would be greatly appreciated!

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# Does this assumption cause problems in (many) cases?

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