Matlab Lab Notes and Answers

7/23/2019 Matlab Lab Notes and Answers

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APPLICATION OF DIFFERENTIAL EQUATIONS

FALLING BODIES

Q1:

Rewrite the second order function in terms of y1 y! see your

notes on th"tLet Y1=xY2=y1=dx_dtY2= d2x_dt2Then you will need to create 2 m-files a follow

Diffential Equation Codesfunctiondh_dt=diff_fun(t,y)lo!alc mdh_dt="y(2)#c$x(2)%m-


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end

Main codesfunctiondiff_mainlo!alm c m='#=*+#c=*1#otions=odeset(.elTol,1e-/)#tsan=",1

y="1#"t,x&=ode0'(diff_fun,tsan,y,otions)heiht= x(end,1) this reresents heiht at 1seccalculate chane in heihtdh= radient (x(3,1))#h_1= dh(end)#calculate chane in timedt= diff(t)#

t_1=dt(end)#calculate chane in 4elocity4elocity = h_1%t_1lot fiure to see motionlot(t,x(3,1),r-)#leend(h)#yla!el(heiht "m&)#xla!el(time "s&)

end

#o edit the $ine wei%ht "nd font wei%ht of your &$ot'$ot (t)(:1*+r,-+$inewidth+!*.$"/e$(-time0s++fontwei%ht++/o$d+*


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Q! is simi$"r to 21 "nd see notes on th"t

One e)tr" %$o/"$ f"ctor "dded d


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Q3

Diffential Equation Codesfunctiondx_dt=diff_fun(t,x)lo!alc m 5dx_dt="x(2)#(-c$x(2)-5$x(2)62$cos(2$t))%mend

Main codesfunctiondiff_mainlo!alm c 5otions=odeset(7.elTol7,1e-/)#x="1#0m=1#5=2#c=8#tsan=",'"t,x&=ode0'(diff_fun,tsan,x,otions)fiure#


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hold on#lot(t,x(3,1),7r7)#leend(7h7)#yla!el(7h7)#xla!el(7t7)

90

Diffential Equation Codesfunctiondx_dt=diff_fun(t,h)lo!al. dx_dt=-.:2$s;rt(2$$h(1))#end


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Main codesfunctiondiff_mainlo!al. otions=odeset(7.elTol7,1e-/)#x="1.=*1#=*+#tsan=",2"t,x&=ode0'(diff_fun,tsan,x,otions)fiure#hold on#lot(t,x(3,1),7r7)#leend(7h7)#

yla!el(7h7)#xla!el(7t7)

9'


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Let , then

function dx_dt=ode_fun8(t,x)dx_dt="0'$cos(i%8)#

x(8)#-*+end

otions=odeset(7.elTol7,1e-/)#x="#1#/0'sin(i%8)tsan=",1'"x,y&=ode0'(ode_fun8,tsan,x,otions)#

fiure#hold on#lot(y(3,1),y(3,2),77)#leend(7x7,7y7)#yla!el(7y7)#xla!el(7x7)#

Working with Data (see notes)Q1:

(2) ?sin the o!tained olynomal model, calculate the followin

4alue*


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y(*')=>

m,4$e:

load data1*mat_coeffs = olyfit(x,y,/)#fiurelot(x,y,7o-7)hold onyfit = oly4al(_coeffs,x)#lot(x,yfit,7r-7,7Line@idth7,2)#leend(7


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="1'#1'#1'#1'#8#8#8#8

B="11*8# 1+8+*C#1'*C# 1+2*8# 181*C# 1/02*8# 188*C#1/81Theta="'0*8#'/#1+*C#1+#/0*C#C/*C# 2*8# 2*CT="2# 11*C# 1/*8# 11*8# 80*8# 2*8# 82*C# 2+*8

D="ones(siEe()) lo() lo(B) lo(Theta)y=lo(T)#ce=DFy#

a=ex(ce(1))!=ce(2)c=ce(8)d=ce(0)

1=20*C#B1=18'#T1=1''#

T1=a$(1:!)$(B1:c)$(T1:d)

Gnswers found3a=11*2+11!= 1*0+c= -*/2/d= -*C1+T(20*C,18',1'')=8*'1+0


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98


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8test!

)!9 d"t"(:3*y!9d"t"(:


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x="*2'tsan=",*0"t,x&=ode0'(diff_fun,tsan,x,otions)#fiurehold onlot(t,x(3,1),7r7)#lot(t,x(3,2),77)#leend(7x17,7x27)#yla!el(7x7)#xla!el(7t7)

Q!

functiondx_dt=diff_fun(t,x)dx_dt="x(1)62$x(2)-x(8)#x(1)6x(8)#0$x(1)-0$x(2)-'$x(8)end

%main codeoptions=odeset('RelTol',1e-6);x0=[0,1,1];

tspan=[0,1];[t,x]=ode45(di!!"!#n,tspan,x0,options);

!i$#e&old on


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plot(t,x(,1),'');plot(t,x(,),'$');plot(t,x(,),'*');le$end('x','+','');+la*el('x');xla*el('t')

Q3

functiondy_dx=diff_fun(x,y)dy_dx=x$y$y-y#end

main codeotions=odeset(7.elTol7,1e-/)#x="1tsan=",'"t,x&=ode0'(diff_fun,tsan,x,otions)#fiurehold onlot(t,x(3,1),7r7)#yla!el(7y7)#

xla!el(7x7)


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90

functiondy_dx=diff_fun(t,x)dy_dx="x(1)-x(2)#-x(2)#x(1)$sin(x(8))end

main codeotions=odeset(7.elTol7,1e-/)#x="1,1,1tsan=",1

"t,x&=ode0'(diff_fun,tsan,x,otions)#fiurehold onlot(t,x(3,1),7r7)#lot(t,x(3,2),77)#lot(t,x(3,8),7!7)#leend(7x7,7y7,7E7)#yla!el(7y7)#xla!el(7x7)9'


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functiondy_dt=diff_fun(t,y)dy_dt="y(2)#-8$y(2)-y(1)60$ex(-2$t)-'end

main codeotions=odeset(7.elTol7,1e-/)#


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x="2,-1tsan=",'"t,x&=ode0'(diff_fun,tsan,x,otions)#fiurehold onlot(t,x(3,1),7r7)#lot(t,x(3,2),77)#lot(t,x(3,8),7!7)#leend(7x7,7y7,7E7)#yla!el(7y7)#xla!el(7x7)

9/

Integrals


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IGJK H?.K Y?


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98OO f=(x)(x*:86sin(8$x))#OO ;uad(f,1,2)

90OO f=(x)(ex(2$x)*$sin(x*:0-2$x))OO ;uad(f,-',')

Derivatives!"#! $% C#E"!I&' "& MI*E+ !,E

-&E #-M Q.

91 GM< 2 @.J TQK HGIK @GYMethod 1//0 2345563+345731+3456689//gdiff(0)341

//sdiff(g)341


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orMethod ;//0 2345563+345731+3456689//ggradient(0+341))(;@tanh(34;@>8)34;@>A;47)+09fBero(f+6)Q6 C"& $E -*ED *IE Q8

Errors

;13 Lare 6 small, difference of two num!ers close toether GM

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