Wednesday, March 22, 2023

Solution To Variation Of Parameters Differential Equation

 Solve using variation of parameters:

d2y/dx 2   - dy/ dx  - 2y  =  e 3x

Solution:

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

c  =   e -x +  c 2  e 2x

Assume: 

yp  = 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  1  ,  2   in the complementary function by the respective functions  1 and 2 which  will be determined.  So the resulting function:  11   +  v 2  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:

yp  = 1 e -x +  v 2  e 2x

yp  =  - e 4x e -x /12  +   e x e-2x/3  

=

 - e 3x/12  + e-3x/3  =  e-3x/4 

Yielding the general soln.

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

 

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