Solve the following differential equation using the method o

Solve the following differential equation using the method of undetermined coefficients. x + 3x + 2x = 6sin 2t

Solution

d²x/dt² +3 dx/dt +2x=6sin2t

The general solution of this differential equation is y(t)=Y_c(t) +Y_p(t)

The roots of the characteristic equation m²+3m+2=0 are -1 and -2

Then Y_c(t)= c₁e^-t + c₂e^-2t and

Y_p(t)= A cos2t +B sin2t

Substitute this equation in the given differential equation will help us to find the actual values of A and B.

-4A cos2t-4B sin2t + 3 ( -2A sin2t+2B cos2t) + 2 (A cos2t+B sin2t) = 6sin2t

Equating sine and cosine terms

-2A+6B=0

-6A-2B=6

Solving we get A= -9/10 and B= -3/10

thus Y_p(t)= -9/10 cos2t+ -3/10 sin2t

and the solution is y(t)= Y_c(t)+ Y_p(t).