Math, asked by dhanista, 10 months ago

If Y=sin^-1x then show that (1-x^2)Y5-7xY4-9y3=0​

Answers

Answered by MaheswariS
8

\textbf{Given:}

\mathsf{y=sin^{-1}x}

\textbf{To prove:}

\mathsf{(1-x^2)y_5-7xy_4-9y_3=0}

\textbf{Solution:}

\textsf{Consider,}

\mathsf{y=sin^{-1}x}

\textsf{Differentiate with respect to x}

\mathsf{y_1=\dfrac{1}{\sqrt{1-x^2}}}

\mathsf{{y_1}^2=\dfrac{1}{1-x^2}}

\mathsf{{y_1}^2(1-x^2)=1}

\textsf{Differentiate with respect to x}

\mathsf{{y_1}^2(-2x)+(1-x^2)2y_1y_2=0}

\mathsf{-xy_1+(1-x^2)y_2=0}

\textsf{Differentiate with respect to x}

\mathsf{-(xy_2+y_1)+(1-x^2)y_3+y_2(-2x)=0}

\mathsf{-xy_2-y_1+(1-x^2)y_3-2xy_2=0}

\mathsf{-3xy_2-y_1+(1-x^2)y_3=0}

\textsf{Differentiate with respect to x}

\mathsf{-3(xy_3+y_2)-y_2+(1-x^2)y_4+y_3(-2x)=0}

\mathsf{-3xy_3-3y_2-y_2+(1-x^2)y_4-2xy_3=0}

\mathsf{-5xy_3-4y_2+(1-x^2)y_4-=0}

\textsf{Differentiate with respect to x}

\mathsf{-5(xy_4+y_3)-4y_3+(1-x^2)y_5+y_4(-2x)-=0}

\mathsf{-5xy_4-5y_3-4y_3+(1-x^2)y_5-2xy_4=0}

\mathsf{Rearranging\;terms\;we\;get}

\boxed{\mathsf{(1-x^2)y_5-7xy_4-9y_3=0}}

Answered by mahek77777
15

\textbf{Given:}

\mathsf{y=sin^{-1}x}

\textbf{To prove:}

\mathsf{(1-x^2)y_5-7xy_4-9y_3=0}

\textbf{Solution:}

\textsf{Consider,}

\mathsf{y=sin^{-1}x}

\textsf\red{Differentiate with respect to x}

\mathsf{y_1=\dfrac{1}{\sqrt{1-x^2}}}

\mathsf{{y_1}^2=\dfrac{1}{1-x^2}}

\mathsf{{y_1}^2(1-x^2)=1}

\textsf\red{Differentiate with respect to x}

\mathsf{{y_1}^2(-2x)+(1-x^2)2y_1y_2=0}

\mathsf{-xy_1+(1-x^2)y_2=0}

\textsf\red{Differentiate with respect to x}

\mathsf{-(xy_2+y_1)+(1-x^2)y_3+y_2(-2x)=0}

\mathsf{-xy_2-y_1+(1-x^2)y_3-2xy_2=0}

\mathsf{-3xy_2-y_1+(1-x^2)y_3=0}

\textsf\red{Differentiate with respect to x}

\mathsf{-3(xy_3+y_2)-y_2+(1-x^2)y_4+y_3(-2x)=0}

\mathsf{-3xy_3-3y_2-y_2+(1-x^2)y_4-2xy_3=0}

\mathsf{-5xy_3-4y_2+(1-x^2)y_4-=0}

\textsf\red{Differentiate with respect to x}

\mathsf{-5(xy_4+y_3)-4y_3+(1-x^2)y_5+y_4(-2x)-=0}

\mathsf{-5xy_4-5y_3-4y_3+(1-x^2)y_5-2xy_4=0}

\mathsf\red{Rearranging\;terms\;we\;get}

\boxed{\mathsf{(1-x^2)y_5-7xy_4-9y_3=0}}

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