Home
Class 12
MATHS
If y=A e^(m x)+B e^(n x) , show that (d^...

If `y=A e^(m x)+B e^(n x)` , show that `(d^2y)/(dx^2)-(m+n)(dy)/(dx)+m n y=0` .

Text Solution

Verified by Experts

`y=Ae^(mx)+Be^(nx)`
`implies(dy)/(dx)=Ame^(mx)+"Bne"^(nx)`
`implies(d^(2)y)/(dx^(2))=Am^(2)e^(mx)+"Bn"^(2)e^(nx)`
`L.H.S.=(d^(2)y)/(dx^(2))-(m+n)(dy)/(dx)+mny`
`=[Am^(2)e^(mx)+Bn^(2)e^(nx)]-(m+n)[Ame^(mx)+"Bn"e^(nx)]+mn[Ae^(mx)+Be^(nx)]`
`=Am^(2)e^(mx)+Bn^(2)e^(nx)-"Am"^(2)e^(mx)-"Bmn"e^(nx)-"Amn"e^(mx)-"Bn"^(2)e^(nx)+"Amn"e^(mx)+"Bmn"e^(nx)`
`=0=R.H.S. " " `Hence Proved.
Promotional Banner

Topper's Solved these Questions

  • Continuity and Differentiability

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercies 5.8|6 Videos

  • Continuity and Differentiability

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|23 Videos

  • Continuity and Differentiability

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercies 5.6|11 Videos

  • APPLICATIONS OF INTEGRALS

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|19 Videos

  • DETERMINANTS

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|19 Videos

Similar Questions

Explore conceptually related problems

If y=a e^(2x)+b e^(-x) , show that, (d^2y)/(dx^2)-(dy)/(dx)-2y=0 .

If e^y(x+1)=1 , show that (d^2y)/(dx^2)=((dy)/(dx))^2 .

If e^y(x+1)=1 , show that (d^2y)/(dx^2)=((dy)/(dx))^2 .

If e^y(x+1)=1 , show that (d^2y)/(dx^2)=((dy)/(dx))^2

If e^y(x+1)=1 . Show that (d^2y)/(dx^2)=((dy)/(dx))^2

If y=e^x(sin x+cosx) prove that (d^2y)/(dx^2)-2 (dy)/(dx)+2y=0 .

If y=e^-x cos x , show that (d^2y)/(dx^2)= 2e^-x sin x

If y=Acosn x+Bsinn x , show that (d^2y)/(dx^2)+n^2\ y=0 .

If e^(y) (x+1) =1 show that (d^(2) y)/( dx^(2)) = ((dy)/(dx))^(2)

If y=a e^(m x)+b e^(-m x), then (d^(2y))/(dx^2) is equals to