To integrate linear ordinary differential equations with con

To integrate linear ordinary differential equations with constant coefficients, assume x(t) = exp(st)and obtain the characteristic equation for s. Then, write the general solutions of the following the differential equations: x - 3x + 2x = 0; x + 2x + 10x = 0; x + 36x = 0;

Solution

i)

the characteristic equation is

s^2 -3s + 2 =0

=>

s = 1,2

general solution x(t) = C1e^t + C2 e^2t

ii)

characteristic equation is

s^2 + 2s + 10 = 0

=>

s = -1+3i , -1-3i

the genral solution x(t) = C1e^-t cos(3t) + C2 e^-t sin(3t)

iii)

s^2+36 = 0

s = 6i,-6i

the genral solution is = C1 cos(6t) + C2 sin(6t)