1.

If `y=3 e^(2x)+2 e^(3x)`, prove that `(d^2y)/(dx^2)-5(dy)/(dx)+6y=0`.

Answer» We have
`y=3e^(2x)+2e^(3x)" …(i)"`
`rArr(dy)/(dx)=3.(d)/(dx)(e^(2x))+2.(d)/(dx)(e^(3x))" [on differentiating (i) w.r.t. x]"`
`=(3xx2e^(2x))+(2xx3e^(3x))=(6x^(2x)+6e^(3x))`
`rArr (dy)/(dx)=6(e^(2x)+e^(3x))." ...(ii)"`
On differentiating (ii) w.r.t. x, we get
`(d^(2)y)/(dx^(2))=6.{(d)/(dx)(e^(2x))+(d)/(dx)(e^(3x))}`
`=6.(2e^(2x)+3e^(3x)).`
`therefore((d^(2)y)/(dx^(2))-5(dy)/(dx)+6y)`
`=6(2e^(2x)+3e^(3x))-30(e^(2x)+e^(3x))+(18e^(2x)+12e^(2x))`
`=(12-30+18)e^(2x)+(18-30+12)e^(3x)=0.`
Hence, `((d^(2)y)/(dx^(2))-5(dy)/(dx)+6y)=0.`


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