InterviewSolution
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InterviewSolution
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If `y=(1/2)^(n-1)cos(ncos^(-1)x),`then prove that `y`satisfies the differential equation `(1-x^2)``(d^2y)/(dx^2)-x(dy)/(dx)+n^2y=0` |
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Answer» `y=(1//2^(n-1))cos (n cos^(-1)x)` `therefore" "(dy)/(dx)=-(1)/(2^(n-1))sin(n cos^(-1)x)[(-n)/sqrt((1-x^(2)))]` `"or "(1-x^(2))((dy)/(dx))^(2)=(n^(2))/(2^(2n-2))sin^(2)(n cos^(-1)x)` `=(n^(2))/(2^(2n-2))[1-cos^(2)(n cos^(-1)x)]` `=n^(2)[(1)/(2^(2n-2))-y^(2)]` Differentiating both sides w.r.t x, we get `(1-x^(2))2(dy)/(dx)(d^(2)y)/(dx^(2))-2x((dy)/dx)^(2)=-2n^(2)y(dy)/(dx)` `"or "(1-x^(2))(d^(2)y)/(dx^(2))-x(dy)/(dx)+n^(2)y=0` |
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