Difference between revisions of "Tutorial Week 4"

From UBCMATH WIKI
Jump to: navigation, search
m (Cytryn moved page Tutorial 4 to Tutorial Week 4)
Line 1: Line 1:
 
===Worksheet Questions===
 
===Worksheet Questions===
  
−
#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.
+
#The differential equation
−
## Write down a differential equation describing the system.
+
:$y'' + 2\alpha^2 y' + (\alpha^4-\alpha^2+2\alpha) y = 0$
 +
depends on the parameter $\alpha$ so that its solutions may be qualitatively different for different values of $\alpha$.
 +
##For what values of $\alpha$ does the characteristic equation of the system have real distinct roots?
 +
##For what values of $\alpha$ does the characteristic equation of the system have a real repeated root?
 
##For what values of $\alpha$ does the characteristic equation of the system have complex roots?  
 
##For what values of $\alpha$ does the characteristic equation of the system have complex roots?  
 
## Assuming complex roots, what is the general solution?
 
## Assuming complex roots, what is the general solution?
 
##At what time will the amplitude have decayed to $e^{-1}$ of its original value?
 
##At what time will the amplitude have decayed to $e^{-1}$ of its original value?
 
#For each of the following differential equations, write the general form of its particular solution.  
 
#For each of the following differential equations, write the general form of its particular solution.  
−
##$y''-25y'=2e^{8x}$
+
##$y''-25y'=2e^{8x}+3$
−
##$y''-y=x^2$
+
##$y''-y=x\cos(x)$
 +
##$y''+y=x\cos(x)$
 
##$y''-2y'+y=e^{x}$
 
##$y''-2y'+y=e^{x}$
  
 
===Solutions===
 
===Solutions===
 
#
 
#
−
##$y''+2\alpha^2y'+(\alpha^4-\alpha^2+2\alpha)y=0$ (1 mark)
 
 
##$0<\alpha<2$. 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 $\alpha<2$).
 
##$0<\alpha<2$. 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 $\alpha<2$).
−
##$y(t)=e^{-\alpha^2t}(A\sin\sqrt{\alpha^2-2\alpha}t+B\cos\sqrt{\alpha^2-2\alpha}t)$. 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).  
+
##$y(t)=e^{-\alpha^2t}(A\sin\sqrt{2\alpha-\alpha^2}t+B\cos\sqrt{2\alpha-\alpha^2}t)$. 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).  
 
##$t=\frac{1}{\alpha^2}$ (2 marks - 1 for creating an equation describing amplitude, 1 for solving for the correct $t$)
 
##$t=\frac{1}{\alpha^2}$ (2 marks - 1 for creating an equation describing amplitude, 1 for solving for the correct $t$)
 
#
 
#
 
##$y_c=c_1e^{25x}+c_2$(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)
 
##$y_c=c_1e^{25x}+c_2$(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)
 
##$y_c = c_1e^{-x}+c_2e^x$ (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)
 
##$y_c = c_1e^{-x}+c_2e^x$ (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)
−
##$y_c = c_1e^x+c_2xe^x$ (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)
+
##$y_c = c_1e^x+c_2xe^x$ (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)

Revision as of 23:40, 24 January 2017

Worksheet Questions

  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. $y''-2y'+y=e^{x}$

Solutions

    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 $t$)
    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)
Retrieved from "https://wiki.math.ubc.ca/mathbook/M256/index.php?title=Tutorial_Week_4&oldid=1347"