Question 7 Your answer is INCORRECT Find the homogeneous equ

Question 7 Your answer is INCORRECT Find the homogeneous equation with constant coefficients that has the given general solution y(4) + 8 y \"-4y,\" _ 16 y \' + 16y=0 d) y4+4 y -16 16-0 e)y+8y-4y\" 16y +16-0 f) None of the above

Solution

y=C₁e^2x +C₂xe^2x +C₃cos(2x)+C₄sin(2x)

y\'=2C₁e^2x +C₂e^2x+2C₂xe^2x -2C₃sin(2x)+2C₄cos(2x)

y\'\'=4C₁e^2x +2C₂e^2x+2C₂e^2x +4C₂xe^2x-4C₃cos(2x)-4C₄sin(2x)

y\"=4C₁e^2x +4C₂e^2x +4C₂xe^2x-4C₃cos(2x)-4C₄sin(2x)

y\'\'\'=8C₁e^2x +4C₂e^2x+4C₂e^2x +4C₂e^2x+8C₂xe^2x+8C₃sin(2x)-8C₄cos(2x)

y\'\'\'=8C₁e^2x +12C₂e^2x+8C₂xe^2x+8C₃sin(2x)-8C₄cos(2x)

y⁽⁴⁾=16C₁e^2x +24C₂e^2x++8C₂e^2x+16C₂xe^2x+16C₃cos(2x)+16C₄sin(2x)

y⁽⁴⁾=16C₁e^2x +32C₂e^2x+16C₂xe^2x+16C₃cos(2x)+16C₄sin(2x)

16,-8,-4,2,1

y⁽⁴⁾+8y\" -4y\"\' -16y\' +16y =0 satisfies the above 4 equations

so option A is answer