Show that y= f(x) is a solution for given equations under, please help?

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Answer 1 · 2018-02-27T19:11:11

(a)

Given: #f(x)=sin(2x)#

Compute the #f''(x)#

#f''(x)=-4sin(2x)#

Substitute #y'' = f''(x)# and #y = f(x)# into the equation:

#y'' + 4y = 0#

#-4sin(2x) + 4sin(2x) = 0#

#0=0# verified to be a solution.

(b)

Given #f(x) = xe^(-2x)#

Compute #f'(x)#

#f'(x) = (1-2x)e^(-2x)#

Compute #f''(x)#

#f''(x) = (4x-4)e^(-2x)#

Substitute #y'' = f''(x)#, #y' = f'(x)#, and #y = f(x)# into the equation:

#y'' + 4y' + 4y = 0#

#(4x-4)e^(-2x) + 4((1-2x)e^(-2x)) + 4(xe^(-2x)) = 0#

#0 = 0# verified to be a solution.