Differential equations viewed rigorously.

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When I first encountered differential equations,solving them seemed to be somewhat mechanical and I could not enjoy any taste of it.But after undergoing linear algebra course,I think there is more to understand in it rather that differentiating and integrating.I now want to revisit differential equation again and study it with a rigorous approach by finding reasons behind every subtle thing. First I took in hand the linear differential equations which is quite easy to understand in view of linear algebra,treating the differentiation as a linear operator on $C^\infty$.But there were many more things like exact differential equations,variable separable and homogeneous equations and some of those methods of solving involved partial derivatives.Since I have yet not studied Riemann integral and multivariable calculus,I find it difficult to analyze these situations.Can anyone help me by suggesting a good reference that explains the reason behind every subtle thing?