Tutorial Week 4

Worksheet Questions

This worksheet is not to be handed in. It should be considered a review/practice sheet for the previous week's material in preparation for the midterm tomorrow. For review/practice of earlier material, look at the old midterms posted on the course site ("solutions" on the menu) and previous worksheets.

1. The differential equation $y'' + 2\alpha^2 y' + (\alpha^4-\alpha^2+2\alpha) y = 0$ depends on the parameter $\alpha$ so that its solutions may be qualitatively different for different values of $\alpha$.
   1. For what values of $\alpha$ does the characteristic equation of the system have real distinct roots?
   2. For what values of $\alpha$ does the characteristic equation of the system have a real repeated root?
   3. For what values of $\alpha$ does the characteristic equation of the system have complex roots?
   4. Assuming complex roots, what is the general solution?
   5. 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. $y''-25y'=2e^{8x}+3$
   2. $y''-y=x\cos(x)$
   3. $y''+y=x\cos(x)$
   4. $y''-2y'+y=e^{x}$
3. Find the general solution to the equation $y''+9y=\sin(3t)$.

Solutions

1. 1. $0<\alpha<2$. 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 $\alpha<2$).
   2. $y(t)=e^{-\alpha^2t}(A\sin\sqrt{2\alpha-\alpha^2}t+B\cos\sqrt{2\alpha-\alpha^2}t)$. 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).
   3. $t=\frac{1}{\alpha^2}$ (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) , $y(x) = c_1e^x+c_2xe^x+\frac{1}{2}x^2e^x$ (1 mark)