Tutorial Week 4

Worksheet Questions

1. An adjustable mass-spring system has a damping coefficient and spring constant that can be changed with the parameters $\alpha$. The spring constant is $\alpha^4-\alpha^2+2\alpha$, the damping coefficient is $2\alpha^2$, and a mass of 1 kg is attached to the spring. No external forces are applied to the system.
   1. Write down a differential equation describing the system.
   2. For what values of $\alpha$ does the characteristic equation of the system have complex roots?
   3. Assuming complex roots, what is the general solution?
   4. At what time will the amplitude have decayed to $e^{-1}$ of its original value?
2. For each of the following differential equations, write the general form of its particular solution.
   1. $y''-25y'=2e^{8x}$
   2. $y''-y=x^2$
   3. $y''-2y'+y=e^{x}$

Solutions

1. 1. $y''+2\alpha^2y'+(\alpha^4-\alpha^2+2\alpha)y=0$ (1 mark)
   2. $0<\alpha<2$. Here, they need to write the characteristic equation of the form $ar^2+br+c=0$ (1 mark), remember that complex roots occur when $ b^2-4ac<0$ (1 mark), and then solve for the variable $\alpha$( 2 marks - 1 if they only say $\alpha<2$).
   3. $y(t)=e^{-\alpha^2t}(A\sin\sqrt{\alpha^2-2\alpha}t+B\cos\sqrt{\alpha^2-2\alpha}t)$. Here they need to find that the roots of the characteristic equation (1 mark) simplify to $-\alpha^2\pm\sqrt{\alpha^2-2\alpha}$ and then plug the real and imaginary parts into the formula correctly (2 marks - 1 for using correct form of general equation and 1 for putting real and imaginary parts in the correct places).
   4. $t=\frac{1}{\alpha^2}$ (2 marks - 1 for creating an equation describing amplitude, 1 for solving for the correct $t$)
2. 1. $y_c=c_1e^{25x}+c_2$(1 mark), $y_p=Ae^{8x}$ where $A=\frac{-1}{68}$(2 marks - 1 for general $y_p$ and 1 for solving for constant), $y(x) = c_1e^{25x}+c_2+\frac{-1}{68}e^{8x}$ (1 mark)
   2. $y_c = c_1e^{-x}+c_2e^x$ (1 mark), $y_p =Ax^2+Bx+C$ where $A=-1$, $B=0$, and $C=-2$(2 marks - 1 for general $y_p$ and 1 for solving for constants), $y(x) = c_1e^{-x}+c_2e^x-x^2-2$ (1 mark)
   3. $y_c = c_1e^x+c_2xe^x$ (1 mark), $y_p = Ax^2e^x$ where $A=\frac{1}{2}$ (2 marks - 1 for general $y_p$ and 1 for solving for constant) , $y(x) = c_1e^x+c_2te^x+\frac{1}{2}x^2e^x$ (1 mark)