Question

(D² – 3D + 2)y = e2×(1 + e2x)¯ - ...


Obtain the general solution unless otherwise instructed

Transcribed: (D² – 3D + 2)y = e2*(1 + e2x)¯* -

Answer

Given:

D2-3D+2y=e2x(1+e2x)-1

Formula used:
y=yh+yp

D2-3D+2y=e2x(1+e2x)-1

A linear non-homogeneous ODE with constant coefficient has the form of anDn+......+a1D+a0y=g(x)

The general solution to the above expression can be written as follows,

y=yh+yp

Where,

  • yh is the solution to the homogeneous ODE anDn+......+a1D+a0y=0
  • yp the particular solution is any function that satisfies the non-homogeneous equation

On finding yh by solving (D2-3D+2)y=0

y=c1e2x+c2ex

On finding yp that satisfies (D2-3D+2)y=e2x(1+e2x)-1

yp=e2x-12ln1+e2x+x-ex arctan ex

Therefore the general solution y=yh+yp is y=c1e2x+c2ex+e2x-12ln1+e2x+x-ex arctan ex

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