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Making Equations Separable Many difierential equations that are not separable can be made separable by making a proper substitution. One erample is the class of first-order equations with right-hand sides that are functions of the combination $y/t$ (or $t/y)$. Given such a $DE$ $dtdy?=f(ty?),$ called Euler-homogeneous. $_{4}$ let $v=y/t$. By the productrule, we deducefrom $y=vt$ that $dtdy?=v+tdtdv?,$ so the equation becomes $v+tdtdv?=f(v)$ which separates into $tdt?=f(v)?vdv?$
Apply this method to solve the Euler-homogeneous DEs and IVPs in Problems 41-44. Plot sample solutions on a direction field and discuss. 41. $dtdy?=ty+t?$

Given differential equation dydt=y+tt=yt+1____(1)Here f(yt)=yt+1by Euler homogeneous method, let v=yt?y=vt