1.

यदि ` y= e ^(x) (sin x +cos x ),` सिद्ध कीजिए की` (d^(2)y)/(dx^(2) )-2 (dy)/(dx) +2y =0`

Answer» यहाँ ` y= e^(x) (sinx +cos x ) `
x के सापेक्ष अवकलन करने पर
`(dy)/(dx) =e^(x) *(d)/(dx) (sin x +cos x )+ (sin x +cos x )(d)/(dx)e^(x)`
` " "e^(x)(cos x- sin x)+ (sin x +cos x )*e^(x)`
पुनः अवकलन करने पर
` (d^(2)y) /(dx^(2))=2*(d)/(dx) (e^(x) cos x ) `
` " =2 { e^(x) *(d)/(dx) (cos x )+ cos x (d)/(dx) (e^(x))}`
` " "=2 *(e^(x) (-sin x )+ (cos x ) e^(x) } `
` " "= 2e^(x) (cos x -sin x )`
अब
` ((d^(2)y)/(dx^(2))-2 (dy)/(dx) +2y )=2e^(x) (cos x -sin x ) -4e ^(x) cos x + 2e ^(x) (sin x +cos x )=0`


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