Difference between revisions of "Tutorial Week 4"
From UBCMATH WIKI
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##$y''-2y'+y=e^{x}$ | ##$y''-2y'+y=e^{x}$ | ||
#Find the general solution to the equation $y''+9y=\sin(3t)$. | #Find the general solution to the equation $y''+9y=\sin(3t)$. | ||
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===Solutions=== | ===Solutions=== | ||
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##$y_c = c_1e^{-x}+c_2e^x$ (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) , $y(x) = c_1e^x+c_2xe^x+\frac{1}{2}x^2e^x$ (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) | ||
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Revision as of 00:20, 28 January 2017
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.
- 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 $\alpha$.
- 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?
- Assuming complex roots, what is the general solution?
- 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.
- $y''-25y'=2e^{8x}+3$
- $y''-y=x\cos(x)$
- $y''+y=x\cos(x)$
- $y''-2y'+y=e^{x}$
- Find the general solution to the equation $y''+9y=\sin(3t)$.
