MATH 308 Lecture 9

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End Exam 1 content

Lecture Notes

Exact Equations

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 \left( \frac{\sin{y}}{y} - 2 \mathrm{e}^{-x} \, \sin{x} \right) \, \mathrm{d}x + \left( \frac{\cos{y} + 2 \mathrm{e}^{-x} \, \cos{x}}{y} \right) \, \mathrm{d}y = 0}

Recall 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 \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}} in order for the differential equation to be considered exact

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} \frac{\partial M}{\partial y} &= \frac{\cos{y}}{y} - \frac{\sin{y}}{y^2} \\ \frac{\partial N}{\partial x} &= \frac{2}{y} \left( -\mathrm{e}^{-x} \, \cos{x} - \sin{x} \, \mathrm{e}^{-x} \right) \end{align}}

Equation is not exact so we can stop here.

We multiply by the integrating factor 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 \mu(x,y) = y\, \mathrm{e}^x} to obtain

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 \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} = \cos{y} \, \mathrm{e}^{x} - 2 \sin{x}}

Now we differentiate 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 F(x,y) = \left( \sin{y} \, \mathrm{e}^x - 2y \, \sin{x} \right) \, \mathrm{d}x + \left( \cos{y}\,\mathrm{e}^x + 2 \cos{x} \right) \, \mathrm{d}x = 0} with respect to $y$

${\begin{aligned}{\frac {\partial F}{\partial y}}&=\cos {y}\,\mathrm {e} ^{x}+2\cos {x}+k'(y)\\\ldots \end{aligned}}$

Implicit solution: $\sin {y}\,\mathrm {e} ^{x}+2y\,\cos {x}+C=0$

$y'(t)=f(y)=\mathrm {e} ^{y}-1$

Equilibrium solution:

${\begin{aligned}f(y)&=0\\\mathrm {e} ^{y}-1&=0\\\mathrm {e} ^{y}&=1\\y&=\ln {1}\\y&=0\end{aligned}}$

Is it stable/unstable/semi-stable?

• for $y<0$, the function is decreasing.
• for $y>0$, the function is increasing.
• unstable.

Phase line:

 -----------<-----------|----------->-----------
          y < 0         0         y > 0
         y' < 0 (dec)            y' > 0 (inc)
        y'' < 0 (ccdn)          y'' > 0 (ccup) 

Concavity?

${\begin{aligned}y'&=f(y)\\y''&=f'(y)\,f(y)\\f'(y)&=\mathrm {e} ^{y}>0\end{aligned}}$

Linear vs Nonlinear

Linear	non-linear
$y'+P(x)y=g(x)\quad y(x_{0})=y_{0}$	$y'=f(x,y)\quad y(x_{0})=y_{0}$
$P(x)$ 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(x)} 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} 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 x_0 \in I} , 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 \in \mathbb{R}}	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 f(x,y)} 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 \tfrac{\partial f}{\partial y}} are continuous on 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 \times J} (product of 2 intervals) 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 x_0 \in I} , 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 \in J}
solution to init. val. problem exists solution is unique solution exists on 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}	solution to init. val. problem exists solution is unique solution exists on interval included in 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}

Exercise 13

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) = \frac{t-y}{2t+5y} \quad y(t_0) = y_0}

nonlinear, 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 f(t,y) = \frac{t-y}{2t+5y}} exists on 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 2t+5y \ne 0}

Solution does not exist on line 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{2}{5} \, t}

Exercise 12

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-3)y' + y \, \ln{t} = 2t \quad y(1) = 2}

Linear diff eq. in "standard" 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' + \frac{\ln{t}}{t-3} \, y = 2t}

Function 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(t) = \frac{\ln{t}}{t-3}} exists on 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 \ne 3} 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 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 t \in \left( 0, 3 \right) \cup \left( 3, \infty \right)}

There exists a unique 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(1) = 2} and the domain of the solution is on (0, 3).