MATH 308 Lecture 12

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Lecture Notes

Theorem of Existence and Uniqueness

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}

Only one solution

Exercise 8

Find the largest interval on which the solution exists.

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} t(t-1)y'' + 3t\,y' + 4y &= 2 & y(-1) &= 0 & y'(-1) &= 3 \\ y'' + \frac{3}{t-1} \, y' + \frac{4}{t(t-1)} \, y &= \frac{2}{t(t-1)} & t &\not\in \{ 0, 1 \} \end{align}}

Since the function is not continuous 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 t = t_0 = 0} , the theorem does not apply.

Exercise 9

Can 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 = \sin{t^2}} be a solution to a second order homogeneous linear equation on an interval containing 0?

(i.e. 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) = 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 \begin{align} y &= \sin{t^2} \\ y' &= 2t\,\cos{t^2} \\ y''&= -4t^2\,\sin{t^2} + 2\cos{t^2} y'' + p(t) \, y' + q(t) \, y &= 0 \end{align}}

Since the function is homogeneous, it has 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(t) = 0} . Assuming 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} and $q$ are continuous for all real numbers, the theorem applies and states that there will be only one unique solution.

Therefore $\sin {t^{2}}$ cannot be a solution.

Note: The constant 0 function is a solution to any homogeneous linear differential equation.

Wronskian

Applies to homogeneous equation; may have non-constant coefficients.

From the beginning of the section, we found that if $y_{1}$ and $y_{2}$ are solutions, then any linear combination $c_{1}\,y_{1}+c_{2}\,y_{2}$ is a solution.

$W(y_{1},y_{2},\ldots ,y_{n})={\begin{vmatrix}y_{1}&y_{2}&\ldots &y_{n}\\{y_{1}}'&{y_{2}}'&\ldots &{y_{n}}'\\\vdots &\vdots &\ddots &\vdots \\{y_{1}}^{(n-1)}&{y_{2}}^{(n-1)}&\ldots &{y_{n}}^{(n-1)}\end{vmatrix}}$

Exercise 10

$y''+y=0$

$y_{1}=\sin {x}$ is a solution, and $y_{2}=\cos {\left(x+{\frac {\pi }{2}}\right)}$ is also a solution. Therefore, $y=c_{1}\,\sin {x}+c_{2}\,\cos {\left(x+{\frac {\pi }{2}}\right)}$ is a general solution.

For the initial value problem $y(0)=1$, $y'(0)=0$, the theorem of existence and uniqueness implies that there exists a (unique) solution $Y$ to the initial value problem. However, $y(0)=0\neq 1$, which is impossible!

The theorem guarantees that $Y$ exists.

Notice that $\cos {\left(x+{\frac {\pi }{2}}\right)}=-\sin {x}$, so our $y$ is essentially the function $y=c_{1}\,\sin {x}-c_{2}\,\sin {x}=c_{3}\,\sin {x}$.

What about $y=c_{1}\,\sin {x}+c_{2}\cos {x}$? This satisfies the initial value problem with $c_{1}=0$ and $c_{2}=1$.

${\begin{aligned}W(\sin {x},\cos {\left(x+{\frac {\pi }{2}}\right)}&=\ldots =-\cos {\left(x-\left(x+{\frac {\pi }{2}}\right)\right)}=0\\W(\sin {x},\cos {x})&=-1\end{aligned}}$

Theorem

Suppose 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} 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} are two solutions of the 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 y'' + p(t) \, y' + q(t) \, y = 0}

For any 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 y(t_0) = y_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'(t_0) = y'_0} , there exists two constants 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} 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} such 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 = c_1 \, y_1 + c_2 \, y_2} is a solution to the initial value problem iff 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 W(y_1, y_2) \ne 0} 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 t_0} .

The solutions to the differential equation is the set of functions

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 \, y_1 + c_2 \, y_2}

iff 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 W(y_1, y_2) \ne 0} 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 t_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_1} 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} are said to form a fundamental set of solutions 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 W(y_1, y_2) \ne 0}

Abel's Theorem

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 y_1} 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} are two solutions to the differential 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 y'' + p(t) \, y' + q(t) \, y = 0} , then there exists a constant 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} such 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 W(y_1, y_2) = C \, \exp{\left( - \int p(t) \, \mathrm{d}t \right)}}

Exercise 13

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 y_1} 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} are two solutions of the differential 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 t^2 \, y'' - 2y' + t \, \mathrm{e}^{t} \, y = 0} , and 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 W(y_1, y_2)(2) = 3} , find 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 W(y_1, y_2)(4)} .

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} W(y_1,y_2) = C \, \mathrm{e}^{\int \frac{2}{t^2} \, \mathrm{d}t} = C \, \mathrm{e}^{-\frac{2}{t}} \\ W(y_1,y_2)(2) &= 3 = C \, \mathrm{e}^{\frac{2}{2}} = \frac{C}{\mathrm{e}} \\ C &= 3\mathrm{e} \\ W(y_1,y_2)(4) &= C \, \mathrm{e}^{-\frac{2}{4}} = 3\mathrm{e} \, \mathrm{e}^{-\frac{1}{2}} = 3 \sqrt{\mathrm{e}} \end{align}}