InterviewSolution
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InterviewSolution
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The differential equations , find the particular solution satisfying the given condition:`2x y+y^2-2x^2(dy)/(dx)=0; y=2`when x = 1 |
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Answer» `2xy+y^(2)-2x^(2)(dy)/(dx)=0` `implies (dy)/(dx)=(2xy+y^(2))/(2x^(2))`……….`(1)` It is a homogenous differential equation. Let, `y=vx` `implies (dy)/(dx)=v+x(dv)/(dx)` From equation `(1)` , `v+x(dv)/(dx)=(2x^(2)v+v^(2)x^(2))/(2x^(2))=v+(1)/(2)v^(2)` `implies x(dv)/(dx)=(1)/(2)v^(2)` `implies (2dv)/(v^(2))=(dx)/(x)` `implies 2int(1)/(v^(2))dv=int(dx)/(x)` `implies -(2)/(v)+c=logx` `implies -(2x)/(y)+c=logx` `implies logx+(2x)/(y)=c` .........`(2)` Given, `y=2` at `x=1` `:. log1+(2xx1)/(2)=c` `implies c=1` Therefore, the particular solution of the given equation is `logx+(2x)/(y)=1` |
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