Solve the differential equation, such that the equation passes through the given point (x, y). (Remember to use absolute values where appropriate.)&nbsp;dy/dx=x-6/x, &nbsp; &nbsp; &nbsp;(-1,8)

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Solve the differential equation, such that the equation passes through the given point (x, y). (Remember to use absolute values where appropriate.)&amp;nbsp;dy/dx=x-6/x, &amp;nbsp; &amp;nbsp; &amp;nbsp;(-1,8)

Nick Camelot · Accepted Answer

To solve the given differential equation dydx=x−6x, and find the particular solution that passes through the point (−1,8), we can use separation of variables and integrate both sides.Separating the variables, we have:dyx−6=dxx.Now, let&#039;s integrate both sides:∫dyx−6=∫dxx.Integrating the left side with respect to y and the right side with respect to x, we get:ln|x−6|=ln|x|+C,where C is the constant of integration.Next, we can simplify the equation using logarithmic properties. Taking the exponential of both sides, we have:|x−6|=|x|eC.Since eC is a positive constant, we can rewrite the equation as:x−6=kx or x−6=−kx,where k=eC.Solving these two cases separately:Case 1: x−6=kx,x(1−k)=6,x=61−k.Case 2: x−6=−kx,x(1+k)=6,x=61+k.Now, let&#039;s substitute the given point (−1,8) into the equations and find the corresponding values of k:For Case 1:−1=61−k,1−k=−6,k=7.For Case 2:−1=61+k,1+k=−6,k=−7.Therefore, we have two possible solutions:1. When k=7, the solution is x=61−7=−1.2. When k=−7, the solution is x=61+(−7)=68=34.Hence, the particular solutions of the given differential equation passing through the point (−1,8) are x=−1 and x=34.

Solve the differential equation, such that the equation

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