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InterviewSolution
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अवकल समीकरण `(dy)/(dx)=(x^(2)-y^(2))/(xy)` को हल कीजिए। |
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Answer» `(dy)/(dx)=(x^(2)-y^(2))/(xy)" ….(1)"` यह एक समघातीय अवकल समीकरण है। माना y = vx समीकरण (1) में रखने पर `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)` माना `1-2v^(2)=t` `therefore -4v=(dt)/(dv)` `implies int(dt)/(-4t) =int(dx)/(x)=vdv=(dt)/(-4)` `implies-(1)/(4)logt+logc=logx` `implies logc=logx+log^(t//4)` `implies c=x.t^(1//4)` `impliesc^(4)=x^(4).t=x^(4).(1-2v^(2))` `=x^(4)(1-(2y^(2))/(x^(2)))impliesc_(1)=x^(2)(x^(2)-2y^(2))` |
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