Question

यदि ` y= e^(msin ^(-1))x,` तब दिखाइए की
` " " (1-x^(2)) (d^(2)y)/(dx^(2))-x (dy)/(dx) -m^(2) y=0`

Answer 1

यहाँ ` " "y= e ^(msin ^(-1)x)`
माना ` t= sin ^(-1) x " "rArr y= e^(mt ) `
`rArr " "(dy)/(dt)= me^(mt )=my (1)/(sqrt (1-x^(2)))`
` rArr " "sqrt (1-x ^(2))*(dy)/(dx) =my`
x के सापेक्ष अवकलन करने पर
` (d^2y)/(dx^(2))*sqrt(1-x^(2))+(dy)/(dx)* (1)/(2) (1-x^(2))^(-1//2) (-2 x)= m (dy)/(dx)`
` rArr " "sqrt (1-x^(2))*(d^(2)y)/(dx^(2))-(x)/(sqrt(1-x^(2)))(dy)/(dx)=m (dy)/(dx)`
` rArr (1-x^(2))(d^2y) /(dx^(2))-x (dy)/(dx) =m (dy)/(dx ) sqrt(1-x^(2) )= m^*my`
इसलिए ` " "(1-x^(2))(d^2y)/(dx^(2))-x (dy)/(dx)-m^(2)y=0 `