If `x=sint ,y=sinp t ,"p r o v et h a t"` `(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2y=0.`

Explanation

Given,` x=sin t and y=sin p t`

On differentiating both sides w.r.t. t, we get


`frac{d x}{d t}=cos t text { and } frac{d y}{d t}=cos p t . p `
`
frac{d y}{d x}=frac{frac{d y}{d t}}{frac{d x}{d t}}=frac{cos p t cdot p}{cos t}`


Now on differentiating both sides w.r.t. x, we get
`
frac{d^{2} y}{d x^{2}}=p frac{[cos t(-sin p t, p)-cos p t(-sin t)]}{cos ^{2} t} cdot frac{d t}{d x}`
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