MATH 308 Lecture 3

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Lecture Notes


First Order ODE

Linear

Exercise 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} y'(x) &= \mathrm{e}^x - \sin{x} \\ y(x) &= \mathrm{e}^x + cos{x} + C \end{align}}

"easy case" differential equation: solve for derivative and calculate the antiderivative.


Exercise 2.2

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 x^2 \, y'(x) + 2x\,y(x) = \mathrm{e}^x - \sin{x}}

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 g(x) = x^2} , then we have 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)\,y'(x) + g'(x) \, y(x) = \mathrm{e}^x - \sin{x}}

Note the product rule on the LHS: now we have

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

Exercise 2.2

Note: 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{\mathrm{d} }{\mathrm{d}x} \left( \mathrm{e}^x \, y(x) \right) {{=}} \mathrm{e}^x \left( y'(x) + y(x) \right)}

Exercise 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} y'(x) - y(x)\,\tan{x} &= \sin{x} \, \cos{x} \\ \mu(x) &= \exp(\left(\int -\frac{\sin{x}}{\cos{x}} \, \mathrm{d}x \right) = \cos{x} \\ \cos{x}\,y'(x) - \sin{x}\,y(x) &= \cos^2{x}\,\sin{x} \\ \frac{\mathrm{d}}{\mathrm{d}x} \left( \cos{x}\,y(x) \right) &= \cos^2{x} \, \sin{x} \\ \cos{x}\,y(x) &= \frac{1}{3}\,\cos^3{x} + C \\ y(x) &= \frac{1}{3}\,\cos^2{x} + C\sec{x} \end{align}}

Integrating Factor

For an ODE 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'(x) + P(x)\,y(x) = g(x)}

The integrating factor is defined as

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) = \exp{\left( \int P(x) \,\mathrm{d}x \right)}}

such that the following ODE is solvable by the product rule:

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} \mu(x) \left(y'(x) + P(x) \, y(x) \right) &= \mu(x) \, g(x) \\ \frac{\mathrm{d}}{\mathrm{d}x} \left( \mu(x)\,y(x) \right) &= \mu(x) \, g(x) \\ \mu(x) \, y(x) &= \int \mu(x) \, g(x) \, \mathrm{d}x \\ y(x) &= \frac{1}{\mu(x)} \, \int \mu(x) \, g(x) \, \mathrm{d}x \end{align}}

In Maple, type 

> intfactor(ODE);

Exercise 4

Solve the ODE 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\,y'-2y=\sqrt{x}} given the 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(1) = 0}

Rewrite in 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{2}{x} \, y = \frac{1}{\sqrt{x}}} to find 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) = x^{-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} \frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{y}{x^2} \right) &= x^{-\frac{5}{2}} \\ \frac{y}{x^2} &= \frac{2}{3} \, x^{-\frac{3}{2}} + C \\ \end{align}}

At this point, we plug in our initial conditions to 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} before finding 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} .

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{0}{1} &= -\frac{2}{3} + C \\ C &= \frac{2}{3} \end{align}}

Now we can solve the ODE 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 y} .

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}{3} \, \sqrt{x} + \frac{2}{3} \, x^2}

Exercise 5

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' + \frac{t+1}{t} \, y &= 1 \\ \mu(t) &= \exp{\left(\int 1 + t^{-1} \,\mathrm{d}x \right)} = \mathrm{e}^{t+\ln{t}} \\ \frac{\mathrm{d}}{\mathrm{d}t} \left( \mathrm{e}^{t + \ln{t}} \, y(t) \right) &= \mathrm{e}^{t + \ln{t}} = t \, \mathrm{e}^t \\ t\,\mathrm{e}^t \, y(t) &= t\,\mathrm{e}^t - \mathrm{e}^t + C \\ y(t) &= 1 - t^{-1} + \frac{C}{t \, \mathrm{e}^t} \end{align}}

Given the inital 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(\ln{2}) = 1} (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 t > 0} ), 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(t) = 1-t^{-1} + \frac{2}{t\,\mathrm{e}^t}}

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