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# Linear differential equation

| 🗖 nosplit | ↑ parent "Differential equation" | words: 172 | descendant words: 172 | descendants: 1
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The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a linear polynomial of multiple variables, which by definition must have the super simple form of: $$f(x0​,x1​,...,xn​)=c0​x0​+c1​x1​+...+cn​xn​+k (126)$$ and then we just put the unknown and each derivative into that simple polynomial: $$f(y(x),y′(x),...,y(n)(x))=c0​y+c1​y′+...+cn​y(n)+k (127)$$ except that now the are not just constants, but they can also depend on the argument (but not on or its derivatives).
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Explicit solutions exist for the very specific cases of:
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