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Differential Equations

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Moderate

Techniques for Nonlinear Equations of Homogeneous Type

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Consider an equation of the homogeneous type, $y'=f(\cfrac{ax+by}{cx+dy})$. A common way to solve such an equation is to introduce $u=y/x$.

Use this technique, we can find the solution of $y'=\cfrac{x+y}{x-y}$ that passes through $(1,0)$. This solution can be written beautifully in polar coordinates $r=F(\theta)$.

The function $F(z)$ is given by (we use variable $z$ here because $z$ is easier to type out):