Math 217 Fall 2000 Exam 2Notational Remark: In this exam, the symbol ∂∂x( )y x means dydx.1. Suppose that ( )y x is a solution to the differential equation = − + ∂∂2x2( )y x 4∂∂x( )y x 3 ( )y x 0. Then ( )y x might equal which of the following expressions: a) + 2 exe( )−3 x b) + e( )−x2 e( )3 x c) − 5 exe( )3 x d) − e( )−xe( )−3 x e) + exx ex f) + e( )2 x2 x e( )2 x g) e( )−4 x( )sin 3 x h) e( )4 x( )sin 3 x i) e( )2 x( )sin 3 x j) e( )( )−2 x( )sin 3 xSolution: c> factor(r^2-4*r+3);( ) − r 1 ( ) − r 3> dsolve(diff(y(x),x$2)-4*diff(y(x),x)+3*y(x) = 0,y(x)); = ( )y x + _C1 ex_C2 e( )3x2. Suppose that ( )y x is a solution to the differential equation = + 4∂∂2x2( )y x 100 ( )y x 0. Then ( )y x might equal which ofthe following expressions: a) − 2 ( )cos 2 x 3 ( )sin 2 x b) + 3 ( )sin 2 x x ( )sin 2 x c) 4 ex( ) + ( )cos x ( )sin x d) − e( )10 xe( )4 x e) + e( )5 xe( )10 x f) − ( )cos 5 x 3 ( )sin 5 x g) − ( )cos 10 x 2 ( )sin 10 x h) − ( )cos 100 x 2 ( )sin 100 x i) + 3 ( )cos 400 x ( )sin 400 x j) + 2 ( )cos 20 x ( )sin 20 xSolution: f> dsolve(4*diff(y(x),x$2) + 100*y(x) = 0,y(x)); = ( )y x + _C1 ( )cos 5 x _C2 ( )sin 5 x3. Suppose that ( )y x is a solution of the differential equation = − + ∂∂2x2( )y x 8∂∂x( )y x 25 ( )y x 0. Then ( )y x might be equal to: a) + 7 e( )3 x11 e( )4 x b) + 7 e( )3 x11 e( )5 x c) + 7 e( )4 x11 e( )5 x d) + 7 e( )4 x( )cos 25 x 11 e( )4 x( )sin 25 x e) + 7 e( )3 x( )cos 5 x 11 e( )3 x( )sin 5 xf) + 7 e( )2 x( )cos 3 x 11 e( )2 x( )sin 3 x g) + 7 e( )4 x( )cos 3 x 11 e( )4 x( )sin 3 x h) + 7 e( )5 x( )cos 4 x 11 e( )5 x( )sin 4 x i) + 7 e( )25 x( )cos 8 x 11 e( )25 x( )sin 8 x j) + 7 e( )25 x( )cos 4 x 11 e( )25 x( )sin 4 xSolution: g > dsolve(diff(y(x), x$2)-8*diff(y(x),x)+25*y(x) = 0,y(x)); = ( )y x + _C1 e( )4x( )sin 3 x _C2 e( )4x( )cos 3 x4. Which of the following has a form that is appropriate for a particular solution of the equation = − ∂∂x( )y x ( )y x + x ( )cos 2 x . a) c1x b) c1( )cos 2 x c) + c1x c2( )cos 2 x d) + + c1x c2( )cos 2 x c3( )sin 2 x e) + + + c1c2x c3( )cos 2 x c4( )sin 2 x f) c1ex g) + c1c2ex h) + + c1c2x c3ex i) + c1x ( )cos 2 x c2x ( )sin 2 x j) + + + c1x ( )cos 2 x c2x ( )sin 2 x c3x2( )cos 2 x c4x2( )sin 2 x Solution: e> dsolve(diff(y(x),x)-y(x) = x+cos(2*x),y(x)); = ( )y x − − − + + x 115( )cos 2 x25( )sin 2 x ex_C15. When the Method of Undetermined Coefficients is used to solve the differential equation = + y ( )'' x π ( )y x x2ex( )sin x how many undetermined coefficients must be determined? a) 9 b) 8 c) 7 d) 6 e) 5 f) 4 g) 3 h) 2 i) 1 j) 0 Solution: d > particularSolution := y(x) = sum(x^k*exp(x)*(c[k,1]*cos(x)+c[k,2]*sin(x)),k=0..2);particularSolution ( )y x ex( ) + c,0 1( )cos x c,0 2( )sin x x ex( ) + c,1 1( )cos x c,1 2( )sin x + = := x2ex( ) + c,2 1( )cos x c,2 2( )sin x + > ode := diff(y(x), x$2) + Pi*y(x) = x^2*exp(x)*sin(x); := ode = + ∂∂2x2( )y x π ( )y x x2ex( )sin x> subs(particularSolution,ode):> undeterminedCoefficientsIdentity := simplify(%):> solve(identity(undeterminedCoefficientsIdentity,x),{seq(c[k,1],k=0..2)} union {seq(c[k,2],k=0..2)} ); = c,2 1−21 + π24 = c,0 18π ( )− + 20 3 π2( ) + π24 ( ) + + 8 π216 π4 = c,1 2−4 + − π24 π 4 + + 8 π216 π4, , ,{ = c,0 2−2 − + π448 π248( ) + π24 ( ) + + 8 π216 π4 = c,1 1−4 − − π24 π 4 + + 8 π216 π4 = c,2 2π + π24, , }6. When the Method of Variation of Parameters is used to find a particular solution of the differential equation = + ∂∂2x2( )y x ( )y x ( )sec x which of the following integrals arises:a) d⌠⌡ ( )sec x x b) d⌠⌡( )sec x2x c) d⌠⌡ ( )tan x x d) d⌠⌡ ( )sec x ( )tan x x e) d⌠⌡( )tan x2x f) d⌠⌡ ( )sin x ( )tan x x g) d⌠⌡x ( )tan x x h) d⌠⌡x ( )sec x x i) d⌠⌡( )sin x2( )sec x x j) d⌠⌡x ( )ln ( )sec x xSolution: c > dsolve(diff(y(x),x$2)+y(x) = 0, y(x)); = ( )y x + _C1 ( )sin x _C2 ( )cos x> y1 := x -> cos(x); y2 := x -> sin(x); := y1 cos := y2 sin> with(linalg):> w := wronskian(vector([y1(x),y2(x)]),x); := w( )cos x ( )sin x− ( )sin x ( )cos x> f := x -> sec(x); := f sec> -y1(x)*Int(y2(x)*f(x)/det(w),x) + y2(x)*Int(y1(x)*f(x)/det(w),x);− + ( )cos x d⌠⌡( )sin x ( )sec x + ( )cos x2( )sin x2x ( )sin x d⌠⌡( )cos x ( )sec x + ( )cos x2( )sin x2x> simplify(%);− + ( )cos x d⌠⌡( )sin x( )cos xx ( )sin x d⌠⌡1 x 7. The graph of a function → t ( )x t is plotted below. What might ( )f t be if ( )x t satisfies the differential equation = + ∂∂2t2( )x t 9 ( )x t ( )f t and if ( )x t has the following grapha) ( )cos 3 b) ( )sin 9 c) ( )cos 3 t d) ( )sin 9 t e) e( )3 t f) e( )−3 t g) e( )3 t h) e( )− 3 t i) ( )sin 3 j) ( )cos 3 t Solution: c The natural frequency is 9 or 3. The graph illustrates resonance. This phenomenon occurs in anundamped mechanical system when the sinusoidal driving frequency is the same as the natural frequency. 8. What is the least positive eigenvalue λ of the endpoint problem , , = + y ( )'' x λ ( )y x 0 …
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