Solution To Variation Of Parameters Differential Equation

 Solve using variation of parameters:

d²y/dx ²   - dy/ dx  - 2y  =  e ^3x

Solution:

This is a 2^nd order DE so write the complementary function:

y _c  = c ₁   e ^-x +  c ₂  e ^2x

Assume: 

y_p  = v ₁ e ^-x +  v ₂  e ^2x

Then:

y₁  = e ^-x ,   y ₂  = e ^2x   ,    f (x) = e ^3x

Therefore:

a) v ‘₁ e ^-x +  v’ ₂  e ^2x    = 0

b) v ‘₁  (-e ^-1) +  v’ ₂   (2e ^2x )   = e ^3x

Then we solve the above simultaneous eqns. to get:

 v’₁  = e ^-4x/3 ,   v’₂  = e ^x/3

From which we obtain:

v’₁  = e ^-4x/12                    v’₂  = e ^x/3

The procedure entails replacing the arbitrary constants  c ₁  ,  c ₂   in the complementary function by the respective functions  v ₁ and v ₂ which  will be determined.  So the resulting function:  v ₁ y ₁   +  v ₂  y ₂

will be a particular integral of:  

a _o (x) y”   +   a ₁ (x) y’  + a ₂ (x) y = b(x)

Substitute into the original eqn:

y_p  = v ₁ e ^-x +  v ₂  e ^2x

y_p  =  - e ^4x e ^-x /12  +   e ^x e^-2x/3  

=

 - e ^3x/12  + e^-3x/3  =  e^-3x/4 

Yielding the general soln.

y  =  c ₁   e ^-x +  c ₂  e ^2x +  e^-3x/4