यदि (If) `y=log(1+cosx)`, सिद्ध करें कि (prove that) `(d^(3)y)/(dx^(3))+(d^(2)y)/(dx^(2)).(dy)/(dx)=0`

Explanation

दिया है, `y=log(1+cosx)" "...(1)`
`implies" "(dy)/(dx)=(1)/(1+cosx)(-sinx)=(-sinx)/(1+cosx)" "...(2)`
`implies" "(d^(2)y)/(dx^(2))=-((1+cosx)cdot cosx-sinx(-sinx))/((1+cosx)^(2))`
`=-(cosx+cos^(2)x+sin^(2)x)/((1+cosx)^(2))`
`=-(cosx+1)/((1+cosx)^(2))=(-1)/(1+cosx)" "...(3)`
`implies" "(d^(3)y)/(dx^(3))=(-1)(-1)(1+cosx)^(-2)(-sinx)`
`=(-sinx)/((1+cosx)^(2))" "...(4)`
अब `(d^(3)y)/(dx^(3))+(d^(2)y)/(dx^(2)).(dy)/(dx)`
`=(-sinx)/((1+cosx)^(2))+(1)/(1+cosx)xx(-sinx)/(1+cosx)" "[(2), (3)" और "(4)" से"]`
`=(-sinx+sinx)/((1+cosx)^(2))=0`