Differential Equations | Part IA, 2004

Construct the general solution of the system of equations

x˙+4x+3y=0y˙+4y3x=0\begin{aligned} &\dot{x}+4 x+3 y=0 \\ &\dot{y}+4 y-3 x=0 \end{aligned}

in the form

(x(t)y(t))=x=j=12ajx(j)eλjt\left(\begin{array}{l} x(t) \\ y(t) \end{array}\right)=\mathbf{x}=\sum_{j=1}^{2} a_{j} \mathbf{x}^{(j)} e^{\lambda_{j} t}

and find the eigenvectors x(j)\mathbf{x}^{(j)} and eigenvalues λj\lambda_{j}.

Explain what is meant by resonance in a forced system of linear differential equations.

Consider the forced system

x˙+4x+3y=j=12pjeλjty˙+4y3x=j=12qjeλjt\begin{aligned} &\dot{x}+4 x+3 y=\sum_{j=1}^{2} p_{j} e^{\lambda_{j} t} \\ &\dot{y}+4 y-3 x=\sum_{j=1}^{2} q_{j} e^{\lambda_{j} t} \end{aligned}

Find conditions on pjp_{j} and qj(j=1,2)q_{j}(j=1,2) such that there is no resonant response to the forcing.

Typos? Please submit corrections to this page on GitHub.