Math 217 Fall 2000 Exam Suppose that y( x ) is a solution to the differential equation. equal which of the following expressions:

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1 Math 7 Fall 000 Exam Notatioal Remark: I this exam, the symbol x y( x ) meas dy dx.. Suppose that y( x ) is a solutio to the differetial equatio - + = Ł x y ( x ) ł 4 Ł x y( x ) 3 y ( x ) 0. The y ( x ) might ł equal which of the followig expressios: a) e x + e ( -3 x ) b) e ( -x ) + e ( 3 x ) c) 5 e x - e ( 3 x ) d) e ( -x ) - e ( -3 x ) e) e x + x e x f) e ( x ) + x e ( x ) g) e ( -4 x ) si( 3 x ) h) e ( 4 x ) si( 3 x ) i) e ( x ) si( 3 x ) j) e ( ( - ) x ) si( 3 x ) c > factor(r^-4*r+3); ( r ) ( r 3 ) > dsolve(diff(y(x),x$)-4*diff(y(x),x)+3*y(x) = 0,y(x)); y( x ) = _C e x + _C e ( 3 x ). Suppose that y( x ) is a solutio to the differetial equatio 4 y( x ) + 00 y( x ) = 0. The y( x ) might equal which of Ł x ł

2 the followig expressios: a) cos( x ) - 3 si( x ) b) 3 si( x ) + x si( x ) c) 4 e x ( cos( x ) + si( x ) ) d) e ( 0 x ) - e ( 4 x ) e) e ( ) + 5 x e ( ) 0 x f) cos( 5 x ) - 3 si( 5 x ) g) cos( 0 x ) - si( 0 x ) h) cos( 00 x ) - si( 00 x ) i) 3 cos( 400 x ) + si( 400 x ) j) cos( 0 x ) + si( 0 x ) f > dsolve(4*diff(y(x),x$) + 00*y(x) = 0,y(x)); y( x ) = _C cos( 5 x ) + _C si( 5 x ) 3. Suppose that y( x ) is a solutio of the differetial equatio - + = Ł x y ( x ) ł 8 Ł x y( x ) 5 y ( x ) 0. The y( x ) might be ł equal to: a) 7 e ( 3 x) + e ( 4 x ) 7 e ( 3 x) e ( 5 x ) 7 e ( 4 x) e ( 5 x ) 7 e ( 4 x ) cos( 5 x ) e ( 4 x ) si( 5 x ) 7 e ( 3 x ) cos( 5 x ) e ( 3 x ) si( 5 x ) b) + c) + d) + e) +

3 7 e ( x ) cos( 3 x ) e ( x ) si( 3 x ) 7 e ( 4 x ) cos( 3 x ) e ( 4 x ) si( 3 x ) 7 e ( 5 x ) cos( 4 x ) e ( 5 x ) si( 4 x ) 7 e ( 5 x ) cos( 8 x ) e ( 5 x ) si( 8 x ) 7 e ( 5 x ) cos( 4 x ) e ( 5 x ) si( 4 x ) f) + g) + h) + i) + j) + g > dsolve(diff(y(x), x$)-8*diff(y(x),x)+5*y(x) = 0,y(x)); y( x ) = _C e ( 4 x ) si( 3 x ) + _C e ( 4 x ) cos( 3 x ) 4. Which of the followig has a form that is appropriate for a particular solutio of the equatio - = Ł x y( x ) ł y( x ) x + cos( x ). a) c x b) c cos( x ) c) c x + c cos( x ) d) c x + c cos( x ) + c 3 si( x ) e) c + c x + c 3 cos( x ) + c 4 si( x ) f) c e x g) c + c e x h) c + c x + c 3 e x i) c x cos( x ) + c x si( x ) j) c x cos( x ) + c x si( x ) + c 3 x cos( x ) + c 4 x si( x ) e

4 > dsolve(diff(y(x),x)-y(x) = x+cos(*x),y(x)); y( x ) = x cos( x ) + si( x ) + e x _C Whe the Method of Udetermied Coefficiets is used to solve the differetial equatio y ''( x ) + p y( x ) = x e x si( x ) how may udetermied coefficiets must be determied? a) 9 b) 8 c) 7 d) 6 e) 5 f) 4 g) 3 h) i) j) 0 d > particularsolutio := y(x) = sum(x^k*exp(x)*(c[k,]*cos(x)+c[k,]*si(x)),k=0..); particularsolutio := y( x) = e x ( c 0, cos( x ) + c 0, si( x )) + x e x ( c, cos( x ) + c, si( x )) + x e x ( c, cos( x) + c, si( x) ) > ode := diff(y(x), x$) + Pi*y(x) = x^*exp(x)*si(x); ode := y( x) + π y( x ) = x e x si( x) x > subs(particularsolutio,ode): > udetermiedcoefficietsidetity := simplify(%): > solve(idetity(udetermiedcoefficietsidetity,x),{seq(c[k,], k=0..)} uio {seq(c[k,],k=0..)} ); { π ( π ) π + 4 π 4 =, c = π 0, 8, c = + 4 ( π + 4 ) ( 8 π π 4, 4, ) 8 π π 4 c, c, 0 = π 4 48 π + 48 π 4 π 4 π, c = ( π + 4 ) ( 8 π π 4, 4, c = ) 8 π π 4, } π + 4

5 6. Whe the Method of Variatio of Parameters is used to fid a particular solutio of the differetial equatio y( x ) + y( x ) = sec( x ) Ł x ł which of the followig itegrals arises: a) sec d ı ( x ) x b) d ı sec( x ) x c) ta( x ) dx ı d) d ı sec( x ) ta ( x ) x e) ta( x ) dx ı f) d ı si( x ) ta ( x ) x g) d ı x ta( x ) x h) ı x sec( x ) dx i) d ı si( x ) sec( x ) x j) d ı x l ( sec( x ) ) x c > dsolve(diff(y(x),x$)+y(x) = 0, y(x)); y( x ) = _C si( x ) + _C cos( x ) > y := x -> cos(x); y := x -> si(x); y := cos y := si > with(lialg): > w := wroskia(vector([y(x),y(x)]),x); > f := x -> sec(x); w := cos( x ) si( x ) si( x ) cos( x ) f := sec

6 > -y(x)*it(y(x)*f(x)/det(w),x) + y(x)*it(y(x)*f(x)/det(w),x); si( x ) sec( x ) cos( x ) sec( x) cos( x ) dx + si( x) dx cos( x) + si( x) cos( x ) + si( x ) > simplify(%); si( x ) cos( x) dx + si( x) dx cos( x ) 7. The graph of a fuctio t fi x( t ) is plotted below. What might f( t ) be if x( t ) satisfies the differetial equatio x( t ) + 9 x( t ) = f( t ) ad if x( t ) has the followig graph Ł t ł a) cos( 3 ) b) si( 9 ) c) cos( 3 t ) d) si( 9 t ) e) e ( 3 t) f) e ( -3 t) g) e ( 3 t ) h) e (- 3 t ) i) si( 3 ) j) cos( 3 t ) c The atural frequecy is 9 or 3. The graph illustrates resoace. This pheomeo occurs i a

7 udamped mechaical system whe the siusoidal drivig frequecy is the same as the atural frequecy. 8. What is the least positive eigevalue l of the edpoit problem y ''( x ) + l y( x ) = 0, y( p ) = 0, y( -p ) = 0? a) /6 b) /8 c) /4 d) / e) / f) p g) p h) p i) p j) p 4 c > sol := dsolve(diff(y(x),x$)+lambda*y(x) = 0, y(x)); sol := y( x ) = _C si( λ x ) + _C cos( λ x ) > eq := subs(x = Pi,rhs(sol) = 0); eq := _C si( λ π ) + _C cos( λ π) = 0 > eq := subs(x = -Pi,rhs(sol) = 0); > eq + eq; If the coefficiet is ot 0 the cos( λ π) = 0 or eq := _C si( λ π ) + _C cos( λ π) = 0 _C cos( λ π) = 0 > solve(cos(sqrt(lambda)*pi) = 0, lambda); With a little more aalysis this value ca be show to be the required value Let y( x ) = solutio of the iitial value problem c x deote the power series of the uique

8 y '( x ) = y( x ) + x, y( 0 ) =. What is the value of c? a) b) - c) / d) -/ e) /3 f) -/3 g) /3 h) -/3 i) j) - c > dsolve({diff(y(x),x) = y(x)+x^, y(0) = },y(x),series); y( x ) = + x x x3 8 x4 40 x5 O( x 6 ) 0. Let y( x ) = c x deote the power series of a solutio of the differetial equatio y ''( x ) + x y '( x ) + y( x ) = 0. The the coefficiets satisfy which of the followig recurreces? a) c = + c + b) c = + + c + c) c = - + c + d) c = c + e) c = + c ( + ) f) c = + c + g) c = - + c + h) c = + c +

9 i) c = - + c + j) c = + c ( + ) ( + ) i > with(slode); [ DEdetermie, FPseries, FTseries, cadidate_mpoits, cadidate_poits, hypergeom_formal_sol, hypergeom_series_sol, mhypergeom_series_sol, msparse_series_sol, polyomial_series_sol, ratioal_series_sol ] > FPseries(diff(y(x),x$)+x*diff(y(x),x)+y(x) = 0,y(x),c()); FPSstruct _C 0 + _C x + c( ) x, ( ) c( ) + ( ) c ( ) = > subs(=+,(^-)*c()+(-)*c(-)); ( ( + ) ) c ( + ) + ( + ) c( ) > solve(%,c(+)); c( ) +. For which of the followig differetial equatios is 0 a regular sigular poit? I) x y ''( x ) + y '( x ) + x y( x ) = 0 II) si( x ) y '( x ) y ''( x ) + + x y( x ) = 0 x III) x y( x ) y ''( x ) + x y '( x ) + = x 0

10 a) I oly b) II oly c) III oly d) I, II oly e) I, III oly f) II, III oly g) I, II, ad III h) Noe of them (No choices (i) ad (j).) a 0 is a ordiary poit of equatio II ad although it is a sigular poit of equatio III, it is ot a regular sigular poit sice the coeficiet x of y ( x ) is ot aalytic at x = 0.. Whe we seek a Frobeius series cetered at 0 for a solutio, which of the followig equatios is the idicial equatio of the differetial equatio x y ''( x ) y '( x ) + - y( x ) = 0? a) r r - = r - = 0 0 b) r r - - = 0 c) r r + = 0 d) r r + - = 0 e) f) x r r + - = 0 g) r r - + = 0 h) r r + - x r = 0 i) r - - = 0 j) There is o idicial equatio because 0 is ot a regular sigular poit of the equatio. a

11 The equatio may be rewritte as x x + = x x y( x ) x y( x) x y( x) 0. The idicial equatio is therefore: r r ( r ) + 0 = 0. > with(detools): > idicialeq(x^*diff(y(x),x,x)+x*/*diff(y(x),x)-x*y(x) = 0,x,0,y(x)); x = x 0 3. Which oe of the followig Frobeius series might be a d y dy y solutio of the differetial equatio + + = 0? d x dx 4 x a) y( x ) = y( x ) = c x ( - ) c x ( - ) b) y( x ) = c x Ł - 3 ł c) d) y( x ) = y( x ) = c x Ł - c x Ł + ł ł e) y( x ) = c x Ł + 4 ł f)

12 g) y( x ) = y( x ) = c x Ł + c x Ł + 3 ł 3 4 ł h) y( x ) = c x Ł ł i) j) y( x ) = c x Ł + 5 ł f > ode := diff(y(x),x$)+diff(y(x),x)+y(x)/(4*x^) = 0; ode := y( x ) + + = x x y( x ) 4 y( x ) 0 x > idicialeq(ode,x,0,y(x)); > solve(%); x x + = 0 4, 4. Simplify Γ 7 3 Γ. 3 a) /9 b) /9 c) /3 d) 4/9 e) 5/9

13 f) /3 g) 7/9 h) 8/9 i) /7 j) 8/7 d The followig computatio will use two successive applicatios of the followig idetity > GAMMA(z+) = expad(gamma(z+)); Γ ( z + ) = Γ( z ) z > 'GAMMA(4/3+)' = subs(z=4/3,rhs(%)); Γ 7 = 3 4 Γ Repeat this process, or just do > simplify(gamma(7/3)/gamma(/3)); Suppose that f( x ) = { 0 < x ad x < 0 otherwise If F is the Laplace trasform of f, the what is F()? a) / - Ł e b) - ł Ł e c) / + ł Ł e ł d) + Ł ł e e) e f) e g) - e

14 h) - Ł e ł i) / ( e - ) j) ( e - ) a > with(ittras); [ addtable, fourier, fouriercos, fouriersi, hakel, hilbert, ivfourier, ivhilbert, ivlaplace, ivmelli, laplace, melli, savetable ] > laplace(heaviside(t)-heaviside(t-),t,s); > subs(s=, %); s e ( s ) s + e ( - ) 6. Suppose that a differetiable fuctio x satisfies x( 0 ) =. If X is the Laplace trasform of x ad if X( ) = 3, the what is the Laplace trasform of x ' evaluated at? (Assume that is i the domai of both X ad the Laplace trasform of x '.) a) - b) 0 c) d) e) 3 f) 4 g) 5 h) 6 i) 7 j) Not eough iformatio has bee give to determie the value. g > with(ittras); [ addtable, fourier, fouriercos, fouriersi, hakel, hilbert, ivfourier, ivhilbert, ivlaplace, ivmelli laplace melli savetable,,, ] > laplace(d(x)(t),t,s);

15 s laplace ( x( t), t, s ) x( 0 ) > subs(x(0)=, %); s laplace ( x( t), t, s ) > subs(s=, %); laplace ( x( t), t, ) > subs(laplace(x(t),t,)=3, %); 5 7. Suppose that a differetiable fuctio x satisfies x( 0 ) = 7 ad D( x )( 0 ) = -3. If X is the Laplace trasform of x ad if X( ) = 5, the what is the Laplace trasform of x '' evaluated at? (Assume that is i the domai of X ad the Laplace trasforms of both x ' ad x ''.) a) 4 b) 0 c) 6 d) e) 8 f) 4 g) 7 h) 3 i) 9 j) Not eough iformatio has bee give to determie the value. i > with(ittras); [ addtable, fourier, fouriercos, fouriersi, hakel, hilbert, ivfourier, ivhilbert, ivlaplace, ivmelli, laplace, melli, savetable ] > laplace((d@@)(x)(t),t,s); s ( s laplace ( x( t ), t, s ) x( 0 ) ) D( x )( 0 ) > subs( {x(0) = 7, D(x)(0) = -3}, %); > subs(s=, %); s ( s laplace ( x( t), t, s ) 7) laplace ( x( t), t, ) > subs(laplace(x(t),t,)=5, %); 9

16 8. If X is the Laplace trasform of x ad if 7 s - X( s ) =, the what is ( ) s + 4 s + 3 x t? a) 5 e ( - t ) cos( 3 t ) - 7 e (- t ) si( 3 t ) 7 e ( - t ) cos( 3 t ) 5 e (- t ) si( 3 t ) b) - 7 e ( - t ) cos( 3 t ) e (- t ) si( 3 t ) c) - d) e ( - t ) ( ) - cos 3 t 7 e (- t ) si( 3 t ) 4 e ( - t ) cos( 3 t ) 3 e (- t ) si( 3 t ) e) + 5 e ( t ) cos( 3 t ) 7 e ( t ) si( 3 t ) f) - 7 e ( t ) cos( 3 t ) 5 e ( t ) si( 3 t ) g) - 7 e ( t ) cos( 3 t ) e ( t ) si( 3 t ) h) - i) e ( t ) ( ) - cos 3 t 7 e ( t ) si( 3 t ) 4 e ( t ) cos( 3 t ) 3 e ( t ) si( 3 t ) j) + b > with(ittras); [ addtable, fourier, fouriercos, fouriersi, hakel, hilbert, ivfourier, ivhilbert, ivlaplace, ivmelli, laplace, melli, savetable ]

17 > ivlaplace((7*s-)/(s^+4*s+3),s,t); 7 e ( t ) cos( 3 t ) 5 e ( t ) si( 3 t ) 9. If X is the Laplace trasform of x ad if X( s ) = 3 s s, the what is x( t )? (Time-savig tip: ( s + 3 ) ( s - ) I the partial fractio decompositio of X(s), 0 is the umerator of the summad whose deomiator is s + 3.) a) t e ( -3 t) + 5 e t b) t e ( 3 t) + 5 e (-t ) c) 3 e ( -3 t) + 4 t e t d) 3 e ( 3 t) + 4 t e (-t ) e) t e ( -3 t) + 7 e t f) t e ( 3 t) + 7 e (-t ) g) e ( -3 t) + 7 t e t h) e ( 3 t) + 7 t e (-t ) i) 3 t e ( -3 t) - e t j) 3 t e ( 3 t) + e (-t ) a > with(ittras); [ addtable, fourier, fouriercos, fouriersi, hakel, hilbert, ivfourier, ivhilbert, ivlaplace, ivmelli, laplace, melli, savetable ]

18 > ivlaplace((3*s+43+5*s^)/((s+3)^*(s-)),s,t); t e ( 3 t ) + 5 e t 0. If f is a give fuctio, the it is ofte useful to be able to fid a differetial operator L = a D + a - ) - D( a D + a 0 such that Lf = 0. Which of the followig operators, give i factored form, has this property if f( x ) = x e ( - x ) si( 3 x )? a) D + 5 b) ( D + 5 ) c) D + 4 D + 9 d) ( D + 4 D + 9 ) e) D - 4 D + 9 f) ( D - 4 D + 9 ) g) D + 4 D + 3 h) ( D + 4 D + 3 ) i) D - 4 D + 3 j) ( D - 4 D + 3 ) h For e ( x ) si( 3 x ) we would use > (D - (-+3*I))*(D-(--3*I)); > expad(%); Because of the x we must apply this operator twice. ( D + 3 I ) ( D I ) D + 4 D + 3

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at