MATH 308 Lecture 11

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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

$y=c_{1}\,\mathrm {e} ^{r_{1}\,t}+c_{2}\,\mathrm {e} ^{r_{2}\,t}$

If $r$ values are complex conjugates, then real-valued solution is of form

$y=c_{1}\,\cos {\theta }+c_{2}\,\sin {\theta }$

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

$\mathrm {e} ^{i\,t}=\cos {t}+i\sin {t}$

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 $y=c_{1}\,\mathrm {e} ^{a\,t}\,\cos {(b\,t)}+c_{2}\,\mathrm {e} ^{a\,t}\,\sin {(b\,t)}$

Section 3.4

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

Characteristic equation $r^{2}-4r+4=(r-2)^{2}=0$ has two roots at $r=2$.

Let $y_{1}=\mathrm {e} ^{2t}$ and $y_{2}=t\,\mathrm {e} ^{2t}$

The general solution is $y=c_{1}\,\mathrm {e} ^{2t}+c_{2}\,t\,\mathrm {e} ^{2t}$

Exercise 7a

Find the solution to the initial value problem $y''+y'-2y=0$, where $y(0)=1$ and $y'(0)=3$.

${\begin{aligned}r^{2}+r-2&=0\\r&\in \{-2,1\}\end{aligned}}$

General solution is $y=c_{1}\,\mathrm {e} ^{t}+c_{2}\,\mathrm {e} ^{-2t}$. To find particular solution, we need $y'=c_{1}\,\mathrm {e} ^{t}-2c_{2}\,\mathrm {e} ^{-2t}$

${\begin{aligned}1&=c_{1}+c_{2}\\3&=c_{1}-2c_{2}\\c_{1}&={\frac {5}{3}}\\c_{2}=-{\frac {2}{3}}\end{aligned}}$

So the particular solution is $y={\frac {5}{3}}\,\mathrm {e} ^{t}-{\frac {2}{3}}\,\mathrm {e} ^{-2t}$

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

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'' + 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.

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' \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

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'' + 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 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 = \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} .