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# Symmetry: For $y^{\prime}=f(t,y)$ do the following: $(a)$ Sketch the direction fields and identify visual symmetries. $(b)$Conjecture how these graphical symmetries relate to algebraic symmetries in $f(t,y)$.$y^{\prime}=\frac{y^{2}}{t}$

Solution

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The differential equation

$y^{'}=\dfrac{y^{2}}{t}$

Here we can see the slope $y^{'}$ depends on both the coordinates of axis and for fixed $t$ the value of slope for $\pm y$ is same but there is no symmetric about horizontal axis but sign of $t$ define sign of slope which reflect visual symmetric in directional field across vertical axis. Here you can clearly see from graph there is no solution symmetricity

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