InterviewSolution
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InterviewSolution
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Solve the differential equation : `(dy)/(dx)=(x^(2)-y^(2))/(xy)`. |
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Answer» `(dy)/(dx)=(x^(2)-y^(2))/(xy)`……….`(1)` It is a homogenous differential equation. Let `y=vx` `implies (dy)/(dx)=v+x(dv)/(dx)` Put these values in eq. `(1)` `v+x(dv)/(dx)=(x^(2)-v^(2)x^(2))/(x^(2)v)=(1-v^(2))/(v)` `implies x(dv)/(dx)=(1-v^(2))/(v)-v=(1-2v^(2))/(v)` `implies (v)/(1-2v^(2))dv=(dx)/(x)` `implies int(v)/(1-2v^(2))dv=int(dx)/(x)` Let `1-2v^(2)=t` `implies int(dt)/(-4t)=int(dx)/(x)impliesvdv=(dt)/(-4)` `implies -(1)/(4)logt+logc=logx` `implies logc=logx+logt^(1//4)` `implies c=x*t^(1//4)` `implies c^(4)=x^(4)*t=x^(4)*(1-2v^(2))` `=x^(4)(1-(2y^(2))/(x^(2)))` `implies c_(1)=x^(2)(x^(2)-2y^(2))`. |
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