MATH 308 Lecture 4

From Notes
Jump to navigation Jump to search

« previous | Wednesday, January 23, 2013 | next »


Lecture Notes


Homework Problem 6

Find value of 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} for which the solution to with initial condition remains finite as approaches .

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 P(t) = -1} , so integrating factor 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 \mu(t) = \mathrm{e}^{-t}} .

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} \mathrm{e}^{-t} \, y' - \mathrm{e}^{-t} \, y &= \mathrm{e}^{-t} + 3\mathrm{e}^{-t}\sin{t} \\ \frac{\mathrm{d}}{\mathrm{d}t} \left( \mathrm{e}^{-t} \, y \right) &= \mathrm{e}^{-t} + 3\mathrm{e}^{-t}\sin{t} \\ \mathrm{e}^{-t} \, y &= -\mathrm{e}^{-t} + 3 \int \mathrm{e}^{-t} \, \sin{t} \, \mathrm{d}t + C \\ y(t) &= -1 - \frac{3}{2} \cos{t} - \frac{3}{2} \sin{t} + C \, \mathrm{e}^{t} \end{align}}

sine and cosine are bounded functions, so the 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 y(t)} will be bounded 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 \, \mathrm{e}^t} only when 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 = 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(0) = -1 + \frac{3}{2} \left( -1 \right) = -\frac{5}{2}}


Complex Numbers

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 \underbrace{a}_{\text{real part}} + i \, \underbrace{b}_{\text{imaginary part}}}

Plot on coordinate plane with 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) = (a,b)} ; imaginary number can also be represented as magnitude 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 = \sqrt{a^2 + b^2}} and angle 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 \theta} made with 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} -axis:

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 + i \, b = r \mathrm{e}^{i \, \theta} = r \cos{\theta} + i \, r \sin{\theta}}

Therefore, 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 \sin{t} = \Im \left( \mathrm{e}^{i \, t} \right)}

We can use this to avoid integration by parts, and it will come in handy for Chapter 3.


Separable Functions

Try to separate the variables to each side 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 \frac{\mathrm{d}y}{\mathrm{d}x} = f(x,y)}

Separable if the RHS can be expressed as product of 2 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 g(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 p(y)}

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} \frac{\mathrm{d}y}{\mathrm{d}x} &= \left( y^2 + 1 \right) \, x \\ \frac{\mathrm{d}y}{y^2+1} &= x \, \mathrm{d}x \\ \int \frac{\mathrm{d}y}{y^2+1} &= \int x \, \mathrm{d}x \\ \tan^{-1}{y} &= \frac{x^2}{2} + C \\ y &= \tan{\left( \frac{x^2}{2} + C \right)} \end{align}}

  • The 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 \tan^{-1}{y} = \frac{x^2}{2} + C} is called the implicit solution
  • The 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 = \tan{\left( \frac{x^2}{2} + C \right)}} is the explicit solution

Exercise 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}y}{\mathrm{d}x} &= x \, \mathrm{e}^{y + x^2} \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= x \, \mathrm{e}^y \, \mathrm{e}^{x^2} \\ \int \frac{\mathrm{d}y}{\mathrm{e}^y} &= \int x \, \mathrm{e}^{x^2} \, \mathrm{d}x \\ -\mathrm{e}^{-y} &= -\frac{1}{2} \mathrm{e}^{x^2} - C \\ -y &= \ln{\left( -\frac{1}{2} \mathrm{e}^{x^2} - C \right)} \\ y &= -\ln{\left( -\frac{1}{2} \mathrm{e}^{x^2} - C \right)} \end{align}}

Retrieved from "https://notes.komputerwiz.net/w/index.php?title=MATH_308_Lecture_4&oldid=2871"