According to given equation, we have,

y = Ae^mx + Be^nx

Then,

= Ame^mx + Bne^nx

Now, on again differentiating we get,

= Am²e^mx + Bn²e^nx

= Am²e^mx + Bn²e^nx – (m+n)( Ame^mx + Bne^nx) + mn(Ae^mx + Be^nx)

= Am²e^mx + Bn²e^nx- Am²e^mx - Bmne^nx -Amne^mx - Bn²e^nx + Amne^mx + Bmne^nx

= 0

Hence Proved