Three students were given a differential equation y''+π ^2y=0 to solve.
y′′+π2y=0y''+\pi^2y=0y′′+π2y=0
d2ydx2=y′′=D2y\dfrac{d^2y}{dx^2}=y''=D^2ydx2d2y=y′′=D2y
D2y+π2y=0D^2y+\pi^2y=0D2y+π2y=0
(D2+π2)y=0(D^2+\pi^2)y=0(D2+π2)y=0
Auxiliary equation,
D2+π2=0D^2+\pi^2=0D2+π2=0
D2=−π2D^2=-\pi^2D2=−π2
D=±iπD=\pm i\piD=±iπ
Complimentary function is given by,
y=C1eiπ+C2e−iπy=C_1e^{i\pi}+C_2e^{-i\pi}y=C1eiπ+C2e−iπ
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