First Order ODE/y' + 2 x y = 1

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The first order ODE:

$y' + 2 x y = 1$

has the general solution:

$y = e^{-{x^2} } \ds \int_a^x e^{t^2} \rd t$

where $a$ is an arbitrary constant.


This is a linear first order ODE in the form:

$\dfrac {\d y} {\d x} + \map P x y = \map Q x$


$\map p x = 2 x$
$\map Q x = 1$

From Solution to Linear First Order Ordinary Differential Equation:

$\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$


\(\ds y\) \(=\) \(\ds e^{-\int 2 x \rd x} \int e^{\int 2 x \rd x} \rd x + C\)
\(\ds \) \(=\) \(\ds e^{- {x^2} } \int e^{x^2} \rd x + C\) Primitive of Power

Further work on this is not trivial, as $\ds \int e^{x^2} \rd x$ has no solution in elementary functions.