Find general solution y5y9y5y0Solutionthe characteristic equ

Find general solution y\"\'+5y\"+9y\'+5y=0

Solution

the characteristic equation for this can be given by

r³+5r²+9r+5=0

the solutions for this equation is given by

r=-1,-2+i,-2-i.

((i). If r is a distinct real root, then y = e ^rt is a solution.

(ii). If r = ± µi are distinct complex conjugate roots, then y = e ^t cos µt and y = e ^tsin µt are solutions. )

so now the general solution can be given by

y=c₁e^-t+c₂e^-2tcost+c₃e^-2tsint

where c₁,c₂,c₃ are constants.