How did we put the value of v in the equation and where did -2log|x| in RHS go - Maths - Differential Equations

Question

How did we put the value of v in the equation and where did
-2log|x| in RHS go?

Neha Sethi · Accepted Answer

Dear student
We have v=yxand logv2+v+1-23tan-1v+1232=-2logx+Clogyx2+yx+1-23tan-1yx+1232=-2logx+Clogy2+yx+x2x2-23tan-12y+x2x32=-2logx+Clogy2+yx+x2x2+2logx=23tan-12y+x3x+Clogy2+yx+x2x2+logx2=23tan-12y+x3x+C   ∵alogx=logxalogy2+yx+x2x2×x2=23tan-12y+x3x+C    ∵logx+logy=log(xy)logy2+yx+x2=23tan-12y+x3x+C
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How did we put the value of v in the equation and where did -2log|x| in RHS go?