%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Scientific Word Wrap/Unwrap Version 2.5 % % % % If you are separating the files in this message by hand, you will % % need to identify the file type and place it in the appropriate % % directory. The possible types are: Document, DocAssoc, Other, % % Macro, Style, Graphic, PastedPict, and PlotPict. Extract files % % tagged as Document, DocAssoc, or Other into your TeX source file % % directory. Macro files go into your TeX macros directory. Style % % files are used by Scientific Word and do not need to be extracted. % % Graphic, PastedPict, and PlotPict files should be placed in a % % graphics directory. % % % % Graphic files need to be converted from the text format (this is % % done for e-mail compatability) to the original 8-bit binary format. % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Files included: % % % % "/document/96fex1sol.tex", Document, 4949, 9/22/1997, 17:04:32, "" % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% Start /document/96fex1sol.tex %%%%%%%%%%%%%%%%%%%% %% This document created by Scientific Notebook (R) Version 3.0 \documentclass[12pt,thmsa]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{sw20jart} %TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart} %TCIDATA{Created=Mon Aug 19 14:52:24 1996} %TCIDATA{LastRevised=Mon Sep 22 13:04:30 1997} %TCIDATA{CSTFile=Exam.cst} %TCIDATA{PageSetup=72,72,36,36,0} \input{tcilatex} \begin{document} \section{Solutions to Exam I\qquad 96Fall} \subsection{1.} $y^{\prime }=3(y+2)x^{2}$ ;\qquad \qquad \qquad \qquad $y(2)=8$ \vspace{1pt} $\frac{dy}{dx}=3x^{2}(y+2)$ \vspace{1pt} $\int \frac{dy}{y+2}=\int 3x^{2}dx$ \vspace{1pt} \vspace{1pt}$\ln |y+2|=x^{3}+C$ \vspace{1pt} $y+2=Ce^{x^{3}}$ \vspace{1pt} $y=Ce^{x^{3}}-2$ \vspace{1pt} $8=Ce^{8}-2$ $C=\frac{10}{e^{8}}\qquad \qquad \Rightarrow \qquad \qquad y=10e^{x^{3}-8}-2$ \vspace{1pt} \subsection{2.} $\frac{dy}{dx}+3y=5e^{2x}-6$ ;\qquad \qquad \qquad $y\left( 0\right) =2$ \vspace{1pt} $p\left( x\right) =3\qquad \mu (x)=e^{\int 3dx}=e^{3x}$ \vspace{1pt} $e^{3x}\frac{dy}{dx}+3e^{3x}y=5e^{5x}-6e^{3x}$ \vspace{1pt} $\int \frac{d}{dx}(e^{3x}y)=\int 5e^{5x}-6e^{3x}$ \vspace{1pt} $e^{3x}y=e^{5x}-2e^{3x}+C$ \vspace{1pt} $y=e^{2x}-2+\frac{C}{e^{3x}}$ \vspace{1pt} $2=1-2+C$ \qquad $C=3$ \vspace{1pt} $\Rightarrow \qquad y=e^{2x}-2+\frac{3}{e^{3x}}$ \vspace{1pt} \subsection{3.} $\frac{dy}{dx}-\frac{1}{x}y=\frac{1}{x^{2}}y^{2}$ \vspace{1pt} $\frac{1}{y^{2}}\frac{dy}{dx}-\frac{1}{x}\ast \frac{1}{y}=\frac{1}{x^{2}}$ \vspace{1pt} \qquad \qquad \qquad $v=\frac{1}{y}$ \qquad \qquad \qquad $\frac{dv}{dx}=\frac{-1}{y^{2}}\frac{dy}{dx}$ \vspace{1pt} $\frac{dv}{dx}+\frac{1}{x}v=\frac{-1}{x^{2}}$ \vspace{1pt} \qquad \qquad $p(x)=\frac{1}{x}\qquad \mu (x)=e^{\int \frac{1}{x}dx}=x$ \vspace{1pt} $x\frac{dv}{dx}+v=\frac{-1}{x}$ \vspace{1pt} $\int \frac{d}{dx}(xv)=\int \frac{-1}{x}$ \vspace{1pt} $\Rightarrow \qquad xv=-\ln x+C$ \vspace{1pt} \qquad \qquad $v=\frac{-\ln x+C}{x}$ \newline \qquad $\Rightarrow \qquad y=\frac{-x}{\ln x+C}$ \vspace{1pt} \subsection{4.} $(2y^{2}-9xy)+(3xy-6x^{2})y^{\prime }=0$ \vspace{1pt} $(2y^{2}-9xy)dx+(3xy-6x^{2})dy=0$ \vspace{1pt} Multiplying both sides of the equation by $xy$ \qquad $\Rightarrow (2xy^{3}-9x^{2}y^{2})dx+(3x^{2}y^{2}-6x^{3}y)dy=0$ \vspace{1pt} \qquad \qquad $\frac{\partial M}{\partial y}=6xy^{2}-18x^{2}y=\frac{\partial N}{\partial x}\Longrightarrow $ new equation is exact. \vspace{1pt}$\frac{\partial }{\partial y}F\left( x,y\right) =N=(3x^{2}y^{2}-6x^{3}y)$ so that $\qquad F(x,y)=\int (3x^{2}y^{2}-6x^{3}y)dy=x^{2}y^{3}-3x^{3}y^{2}+f(x)$ \vspace{1pt} $\frac{\partial F\left( x,y\right) }{\partial x}=2xy^{3}-9x^{2}y^{2}+f^{% \prime }(x)=M=2xy^{3}-9x^{2}y^{2}$ \vspace{1pt}$\Longrightarrow f^{\prime }\left( x\right) =0$ \ so that $% f\left( x\right) =k$ \ Hence \qquad \qquad $F\left( x,y\right) =x^{2}y^{3}-3x^{3}y^{2}+k\Longrightarrow $ solution is $x^{2}y^{3}-3x^{3}y^{2}=C$ \vspace{1pt} \subsection{5.} $y^{\prime }=y^{2}\qquad \qquad \ \ \qquad 0=y^{2}\qquad \qquad y=0$ \qquad \qquad \qquad \qquad \qquad $1=y^{2}\qquad \qquad y=1$ , $y=-1$ \qquad \qquad \qquad \qquad \qquad $4=y^{2}\qquad \qquad y=2$ , $y=-2$ \qquad \qquad \qquad \qquad \qquad $9=y^{2}\qquad \qquad y=3$ , $y=-3$ \FRAME{itbpF}{4.7712in}{3.8207in}{0in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.7712in;height 3.8207in;depth 0in;display "USEDEF";function \TEXUX{$1,-1,2,-2,3,-3$};linecolor "gray";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$\int y^{2}dy,1+\int y^{2}dy,2+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$3+\int y^{2}dy,4+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$4+\int y^{2}dy,5+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$6+\int y^{2}dy,7+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$-1+\int y^{2}dy,-2+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$-3+\int y^{2}dy,-4+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$-5+\int y^{2}dy,-6+\int y^{2}dy$};linecolor "blue";linestyle 3;linethickness 1;pointstyle "point";function \TEXUX{$1+\int y^{2}dy,-1+\int y^{2}dy$};linecolor "black";linestyle 1;linethickness 2;pointstyle "point";xmin "-5";xmax "5";xviewmin "-3.2";xviewmax "3.2";yviewmin "-4";yviewmax "4";viewset"XY";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{y};var1name \TEXUX{$x$};}} \end{document} %%%%%%%%%%%%%%%%%%%%% End /document/96fex1sol.tex %%%%%%%%%%%%%%%%%%%%%