How you solve this differential equation? #(dphi)/dx+(8pi^2mE)/h^2phi = 0#,where #m,E,h# are constants.

2 Answers
Jan 10, 2018

The solution is #phi=C_1e^(-x((8pi^2mE)/h^2))#

Explanation:

The differential equation is

#(dphi)/dx+kphi=0#

where #k=(8pi^2mE)/h^2#

This is first order differential equation with separable variables

#(dphi)/phi=-kdx#

Integrating

#int(dphi)/phi=int-kdx#

#lnphi=-kx+C#

#phi=e^(-kx+C)=C_1e^(-kx)# where #C_1 in RR#

The solution is

#phi=C_1e^(-x((8pi^2mE)/h^2))#

Jan 10, 2018

#phi(x) = Ae^(Bx)# where #B=-(8pi^2mE)/h^2#

Explanation:

#(dphi)/dx +(8pi^2mE)/h^2phi=0#

Let #-B=(8pi^2mE)/h^2#

#therefore (dphi)/dx -Bphi = 0 rArr (dphi)/dx = Bphi rArr int1/phidphi = intBdx#

So #ln|phi| = Bx+c#

So #phi= e^(Bx+c)=Ae^(Bx)#