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text: 'A **first order lineal equation** has the form\n$$\nu'=a(t)u+b(t)\n$$\nwhere $a$ and $b$ are continuous functionsfrom an interval $[\\alpha, \\beta]$ to the real line.\n\n$b$ is called the inhomogeneity of the problem, and the equation where $b=0$ is called the associated homogeneous equation.\n$$\nu'=a(t)u\n$$\n\nA **solution** of a first order linear equation is a $C^1$ function from $[\\alpha, \\beta]$ to the real line such that the equation is satisfied at all times. We will denote the set of solutions of an equation with inhomogeneity $b$ as $\\Sigma_b$, and the solutions of the associated homogeneous system as $\\Sigma_0$.\n\n## Properties of the space of solutions\n$\\Sigma_0$ is a [3w0 vector space]; that is, it satifies the **principle of superposition**: linear combinations of solutions are solutions.\n\n$\\Sigma_b$ is an [-affine_space] parallel to $\\Sigma_0$. That is, it satifies that the difference of any two solutions are in $\\Sigma_0$, and any element in $\\Sigma_0$ plus other element in $\\Sigma_b$ is an element from $\\Sigma_b$.\n\n## First order linear equations of constant coefficients\nOne special kind of linear equations are those in which the coefficients $a$ and $b$ are constant numbers Such linear equations are always resoluble.\n$$\nu' = au+b\n$$\n\nTo solve them, we first have to solve the associated homogeneous equation $u'=au$.\n\nThis has as a solution the functions $ke^{\\int_{t_0}^ta}$ for $k$ constant and $t_0\\in [\\alpha, \\beta]$.\n\nWe can find a concrete solution of the inhomogeneous equation using **variation of coefficients**.\nWe consider as a candidate to a solution the function $u=h\\dot v$, for $h$ a solution of the homogeneous system such as $e^{\\int_{t_0}^ta}$.\n\nThen if we plug $u$ into the equation we find that\n$$\nu'=(hv)'=h'v+hv'=au+b=a(hv)+b\n$$\nSince $h\\in\\Sigma_0$, $h'=ah$, thus\n$$\nv'=bh^{-1}=be^{-\\int_{t_0}^ta}\n$$\nTherefore we can integrate and we arrive to:\n$$\nv=\\int_{t_0}^tbe^{\\int_{t}^sa}ds\n$$\nBy the affinity of $\\Sigma_b$, we can parametrize it by $ke^{\\int_{t_0}^ta}+\\int_{t_0}^tbe^{\\int_{t}^sa}ds$ for $k$ constant.',
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