2y'' + 5y' + 3y = x + cos(x)
Given that the differential equation represents a mass on a spring with friction, qualitatively rate the spring as weak or strong (in relation to friction). Justify your choice based on your solution.
my work: To start, I looked at this equation and reflected it onto the equation: mx''+cx'+kx = F(t) where m is mass, c is the damping constant, k is the spring constant, and F(t) is the external force. In the case of our equation, m = 2, c = 5, k = 3, and F(t) = x+cos(x).
However, I am unsure of how to proceed after this in order to find out whether the spring is weak or strong. Any help is greatly appreciated!
The characteristic equation of homogeneous part $2y''+5y'+3y=0$ is $$2\lambda^2+5\lambda+3=0$$ has roots $\lambda_2=-\dfrac32<-1=\lambda_1<0$ which shows that the system has any oscillation. With initial condition $y_0$ the solution is $$y(x)=y_0\left(3e^{-x}-2e^{-\frac32x}\right)$$