InterviewSolution
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InterviewSolution
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Prove that `x^2-y^2=c(x^2+y^2)^2`is the general solution of differential equation `(x^3-2xy^2)dx=(y^3-3x^2y)dy`, where c is a parameter. |
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Answer» Given, `(x^(3)3xy^(2))dx=(y^(3)-3x^(2)y)dy` `(dy)/(dx)=(x^(3)-3xy^(2))/(y^(3)-3x^(2)y)` `implies v+x(dv)/(dx)=(x^(3)-3x^(3)v^(2))/(v^(3)x^(3)-3x^(3)v)` Let `y=vx` `implies (dy)/(dx)=v+x(dv)/(dx)` `=(1-3v^(2))/(v^(3)-3v)` `implies x(dv)/(dx)=(1-3v^(2))/(v^(3)-3v)-v` `=(1-3v^(2)-v^(4)+3v^(2))/(v^(3)-3v)` `=(1-v^(4))/(v^(3)-3v)` `implies (v^(3)-3v)/(1-v^(4))dv=(dx)/(x)` `implies int(2v(v(2)-3))/(1-v^(4))dv=2int(dx)/(x)+logc` `implies int(t-3)/(1-t^(2))dt=2logx+logc` (Let `v^(2)=timplies2vdv=dt`) `=int{(-1)/(1-t)-(2)/(1+t)}dt=2logx+logc` (Using partial fractions) `implies (log(1-t)-2log(1+t)=log(cx^(2))` `implieslog(1-t)/((1+t)^(2))=logcx^(2))` `implies (1-t)/((1+t)^(2))=cx^(2)` `implies1-v^(2)=cx^(2)(1+v^(2))^(2)` `implies 1-(y^(2))/(x^(2))=cx^(2)(1+(y^(2))/(x^(2)))^(2)` `implies x^(2)-y^(2)=c(x^(2)+y^(2))^(2)` |
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