Sum
Evaluate `(d^4y)/(dx^4)+2(d^2y)/(dx^2)+y=0`
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Solution
put `d/dx=D`
`therefore D^4y+2D^2y+y=0`
`therefore D^4+2D^2+1=0`
Put `D^2=t`
`=> t^2+2t+1=0`
`=> t=-1, -1`
Roots are : D=+i,-i,+i,-i
The complementary solution of given eqn is
`y_c=y_g=(C_1+xC_2)cosx+(C_3+xC_4)sinx`
Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
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