Determine whether the following sets form a fundamental set ofsolutions to the g

Question

Determine whether the following sets form a fundamental set ofsolutions to the given equation (assume values for x where thefunctions are defined), and if so find the fundamentalsolution: a. y1(x) = 4 and y2(x) = 3x with y" =0 b. y1(x) = -3e-2x and y2(x) =e-2x with y"+4y'+4y = 0 Any help is appreciated. Determine whether the following sets form a fundamental set ofsolutions to the given equation (assume values for x where thefunctions are defined), and if so find the fundamentalsolution: a. y1(x) = 4 and y2(x) = 3x with y" =0 b. y1(x) = -3e-2x and y2(x) =e-2x with y"+4y'+4y = 0 Any help is appreciated.

Explanation / Answer

QuestionDetails: Determine whether the following sets form a fundamental set ofsolutions to the given equation (assume values for x where thefunctions are defined), and if so find the fundamentalsolution: a. y1(x) = 4 and y2(x) = 3x with
y" = 0.......................................1
y1(x) = 4 ..........................2
Y1'=0
Y1''=0...HENCE
y1(x) = 4 IS A SOLUTION TO THE GIVEN D.E.
and
y2(x) = 3x ............................3
Y2'=3
Y2''=0...HENCE
y2(x) = 3x IS A SOLUTION TO THE GIVEN D.E.
HENCE THE 2 GIVEN SOLUTIONS FORM A SET OF SOLUTIONS TO THE GIVEND.E.OF
Y''=0
THEY ARE INDEPENDENT SOLUTIONS SINCE
Y=AY1+BY2=A[4]+B[3X] CAN BE ZERO FOR ALLVALUES OF X ONLY IF A=B=0
HENCE THE 2 GIVEN SOLUTIONS FORM A SET OF FUNDAMENTAL
SOLUTIONS TO THE GIVEN D.E.OF ....Y''=0

b. y1(x) = -3e-2x and y2(x) =e-2x with
y"+4y'+4y =0.............................................1
y1(x) = -3e-2x ........................2
Y1'=+6E-2X.....
Y1''=-12E-2X...HENCE
y"+4y'+4y =-12E-2X+4*6E-2X-4*3E-2X =0....OK
y1(x) = -3e-2x ...... IS A SOLUTION TO THEGIVEN D.E.
and
y2(x) = e-2x with
Y2'=-2E-2X.....
Y2''=4E-2X...HENCE
y"+4y'+4y = 4E-2X+4*(-2)E-2X +4E-2X =0....OK
y2(x) = e-2x ...... IS A SOLUTION TO THEGIVEN D.E.
HENCE THE 2 GIVEN SOLUTIONS FORM A SET OF
SOLUTIONS TO THE GIVEN D.E.OF ....y"+4y'+4y =0
BUT THEY ARE NOT INDEPENDENT SOLUTIONS SINCE
Y=AY1+BY2=A[-3E-2X]+B[E-2X]CAN BE ZERO FOR ALL VALUES OF X ONLY EVEN
FOR NON ZERO VALUES OF A AND B ....FOR EX...A=1...B=3
HENCE THE 2 GIVEN SOLUTIONS DO NOT FORM A SET OF FUNDAMENTAL
SOLUTIONS TO THE GIVEN D.E.OF ....y"+4y'+4y =0

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