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| | #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===
| + | [[Tutorial Week 4 Solutions]] |
| − | # Write the characteristic equation of the form $ar^2+br+c=0$. The discriminant is equal to $4\alpha^2 - 8\alpha$, which has zeroes at $0$ and $2$. Thus
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| − | ## $\alpha < 0$ and $\alpha > 2$.
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| − | ## $\alpha = 0$ and $\alpha = 2$.
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| − | ## $0 < \alpha < 2$.
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| − | ## $y(t)=e^{-\alpha^2t}(A\sin\sqrt{2\alpha-\alpha^2}t+B\cos\sqrt{2\alpha-\alpha^2}t)$. The roots of the characteristic equation are equal to to $-\alpha^2\pm i\sqrt{\alpha^2-2\alpha}$, and $y(t)$ is found by using the real and imaginary parts correctly in the formula.
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| − | ##The amplitude is equal to $M(t) = e^{-\alpha^2 t} \sqrt{A^2 + B^2}$. Solving $M(t) / M(0) = e^{-1}$ gives $t=\frac{1}{\alpha^2}$.
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| − | #
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| − | ## $y_c=c_1e^{25x}+c_2$(1 mark), $y_p=Ae^{8x} + Bx$. Can solve to obtain $A=\frac{-1}{68}$, $B = \frac{-3}{25}$. General solution is $y(x) = c_1e^{25x}+c_2+\frac{-1}{68}e^{8x} - \frac{3}{25} x$.
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| − | ## $y_c = c_1e^{-x}+c_2e^x$. $y_p =A\cos x + B x \cos x + C\sin x + Dx \sin x$ where we can solve for $A=0$, $B=-1/2$, and $C=1/2$ and $D=0$. General solution is $y(x) = y_c(x) + y_p(x)$.
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| − | ## Characteristic polynomial has roots $\pm i$, so $y_c = c_1 \cos x + c_2 \sin x$. The particular solution is thus $Ax\cos x + B x^2 \cos x + Cx \sin x + Dx^2 \sin x$. (The higher powers of $x$ appear because the terms $\cos x$ and $sin x$ are already in the homogeneous solution.) Solve to obtain $A=1/4$, $B=0$, $C=0$, $D = 1/4$.
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| − | ## $y_c = c_1e^x+c_2xe^x$. $y_p = Ax^2e^x$ where $A=\frac{1}{2}$, so $y(x) = c_1e^x+c_2xe^x+\frac{1}{2}x^2e^x$.
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| − | # The characteristic equation has roots $\pm 3i$, so $y_c(t) = c_1 \cos(3t) + c_2 \sin(3t)$. The particular solution has form $y_p(t) = At \cos (3t) + Bt \sin (3t)$. We have solve this to obtain $A=-1/6$ and $B=0$. Thus $y(t) = c_1 \cos(3t) + c_2 \sin(3t) - \frac 1 6 t\cos(3t)$.
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Latest revision as of 23:02, 30 December 2020
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 $y''+9y=\sin(3t)$.
Tutorial Week 4 Solutions
