Find the particular solution of the differential equation d2y/dx2 - 4dy/dx + 4y =0 for which the curve of y passes through the point (0,1) and has a stationary point at x=-1
I presume you have determined that the general solution to \(\displaystyle y''- 4y'+ 4y= 0\) is \(\displaystyle y= (C_1x+ C_2)e^{2x}\). \(\displaystyle y(0)= C_2e^0= C_2= 1\) so you have \(\displaystyle y= (C_1x+ 1)e^{2x}\). Saying that the function has a singular point at x= -1 means that y'(-1) is 0: \(\displaystyle y'= C_1e^{2x}+ 2(C_1x+ 1)e^{2x}= e^{2x}(3C_1x+ 2)\) so that \(\displaystyle y'(-1)= e^{-2}(-3C_1+ 2)= 0\). Solve that for \(\displaystyle C_1\).
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