InterviewSolution
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InterviewSolution
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Show that the given differential equation is homogeneous and solve each of them.`(x^2-y^2)dx+2x ydy=0` |
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Answer» `2xydy=(y^(2)-x^(2))dx` `implies (dy)/(dx)=(y^(2)-x^(2))/(2xy)` `implies v+x(dv)/(dx)=(v^(2)x^(2)-x^(2))/(2x^(2)v)` Let `y=vx` `implies (dy)/(dx)=u+(dv)/(dx)` `implies x(dv)/(dx)=(v^(2)-1)/(2v)-v=(-1(1+v^(2)))/(2v)` `implies(2v)/(1+v^(2))dv=-(dx)/(x)` `implies int(2v)/(1+v^(2))dv=-int(dx)/(x)+logc` `implies log(1+v(2))=-logx+logc` `implies 1+v^(2)=(c )/(x)` `impliesx^(2)+v^(2)x^(2)=cx` `implies x^(2)+y^(2)=cx` |
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