MATH 308 Lecture 11

« previous | Monday, February 11, 2013 | next »

Lecture Notes

Announcements:

• Quiz this Friday over section 3.1 and 3.3
• Homework is due Monday

Homogeneous DEs with Constant Coefficients

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\,y'' + b\,y' + c\,y = 0}

Characteristic equation is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\,r^2 + b\,r + c = 0}

General solutions are of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \mathrm{e}^{r_1\,t} + c_2 \, \mathrm{e}^{r_2\,t}}

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} values are complex conjugates, then real-valued solution is of form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \cos{\theta} + c_2 \, \sin{\theta}}

where the cosine and sine come from Euler's formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{i\,t} = \cos{t} + i \sin{t}}

Exercise 5

${\begin{aligned}y''-2y'+5y&=0\\r^{2}-2r+5&=0\\r&=1\pm 2i\\y_{1}&=\mathrm {e} ^{t+2i\,t}\\&=\mathrm {e} ^{t}\left(\cos {(2t)}+i\,\sin {(2t)}\right)\\y_{2}&=\mathrm {e} ^{t-2i\,t}\\&=\mathrm {e} ^{t}\left(\cos {(-2t)}+i\,\sin {(-2t)}\right)\\{\frac {y_{1}+y_{2}}{2}}&=\mathrm {e} ^{t}\cos {2t}\end{aligned}}$

What if the equation has two identical roots?

In general, if $r=a\pm b\,i$ are roots of the characteristic equation, then the general solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \mathrm{e}^{a\,t} \, \cos{(b\,t)} + c_2 \, \mathrm{e}^{a\,t} \, \sin{(b\,t)}}

Section 3.4

$y''-4y'+4y=0$

Characteristic equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^2 - 4r + 4 = (r-2)^2 = 0} has two roots at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 2} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1 = \mathrm{e}^{2t}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2 = t \, \mathrm{e}^{2t}}

The general solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \mathrm{e}^{2t} + c_2 \, t \, \mathrm{e}^{2t}}

Exercise 7a

Find the solution to the initial value problem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' + y' - 2y = 0} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0) = 1} and $y'(0)=3$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} r^2 + r - 2 &= 0 \\ r &\in \{-2, 1 \} \end{align}}

General solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \mathrm{e}^{t} + c_2 \, \mathrm{e}^{-2t}} . To find particular solution, we need Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = c_1 \, \mathrm{e}^{t} - 2 c_2 \, \mathrm{e}^{-2t}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 1 &= c_1 + c_2 \\ 3 &= c_1 - 2c_2 \\ c_1 &= \frac{5}{3} \\ c_2 = -\frac{2}{3} \end{align}}

So the particular solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{5}{3} \, \mathrm{e}^{t} -\frac{2}{3} \, \mathrm{e}^{-2t}}

Exercise 7b

Find the solution to the initial value problem $9y''-12y'+4y=0$, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0) = 2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(0) = -1} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 9r^2 - 12r + 4 &= 0 \\ (3r-2)^2 &= 0 \\ r &\in \left\{ \frac{2}{3}, \frac{2}{3} \right\} \end{align}}

The general solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \mathrm{e}^{\frac{2t}{3}} + c_2 \, t \, \mathrm{e}^{\frac{2t}{3}}}

We can already find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0) = c_1 = 2} , so ...

So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 2 \mathrm{e}^{\frac{2t}{3}} - \frac{7}{3} \, t \, \mathrm{e}^{\frac{2t}{3}}}

Exercise 8

$y''+2y'+2y=0$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(\tfrac{\pi}{4}) = 2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(\tfrac{\pi}{4} = -2} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} r^2 + 2r + 2 &= 0 \\ r &= -1 \pm i \end{align}}

General solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = c_1 \, \mathrm{e}^{-t} \cos{t} + c_2 \, \mathrm{e}^{-t} \sin{t}} .

Find particular solution by plugging in initial conditions:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y \left( \frac{\pi}{4} \right) = 2 &= c_1 \, \mathrm{e}^{-\frac{\pi}{4}} \, \frac{\sqrt{2}}{2} + c_2 \, \mathrm{e}^{-\frac{\pi}{4}} \, \frac{\sqrt{2}}{2} \\ c_1 + c_2 = 2 \sqrt{2} \, \mathrm{e}^{\frac{\pi}{4}} \end{align}}

Differentiate the general solution: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = -c_1 \, \mathrm{e}^{-t} \, \cos{t} - c_1 \, \mathrm{e}^{-t} \, \sin{t} - c_2 \, \mathrm{e}^{-t} \, \sin{t} + c_2 \, \mathrm{e}^{-t} \, \cos{t}}

And plug in initial condition.

$y'\left({\frac {\pi }{2}}\right)=-2c_{1}\,\mathrm {e} ^{-{\frac {\pi }{4}}}\,{\frac {\sqrt {2}}{2}}$

Now we can solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 = \sqrt{2} \mathrm{e}^{\frac{\pi}{4}}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 = \sqrt{2} \mathrm{e}^{\frac{\pi}{4}}} , so the particular solution is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \sqrt{2} \mathrm{e}^{\frac{\pi}{4}} \, \mathrm{e}^{-t} \, \cos{t} + \sqrt{2} \mathrm{e}^{\frac{\pi}{4}} \, \mathrm{e}^{-t} \, \sin{t}}

Theorem of Existence and Uniqueness

Consider the initial value problem

$y''+p(t)\,y'+q(t)\,y=g(t)\quad y(t_{0})=y_{0}\quad y'(t_{0})=y'_{0}$

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are continuous on an open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} that contains the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0} . Then there exists exactly one solution $y=\Phi (t)$ of this problem, and the solution exists throughout the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} .