2. For which values of the parameters α and β has the linear system. dy/dt x + y

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1 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA134 Differential Equations and Transform Methods Date: Write time: hours Aid: Writing materials This examination consists of eight randomly ordered problems each of which is worth at maximum points. The pass-marks 3, 4 and require a minimum of 18, 26 and 34 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 18, 20, 26, 33 and 38 respectively. Solutions are supposed to include rigorous justifications and clear answers. All sheets of solutions must be sorted in the order the problems are given in. 1. At time 0, there are 80 grams of chemical A and 60 grams of chemical B. The chemicals are combined whereby they react together in the proportions 2 : 3 forming the chemical C, i.e. for each grams of the final product C, it will be used 2 grams of A and 3 grams of B. It is assumed that the energy consumed in the reaction is negligible in relation to the energies which the masses represent. The rate of the reaction is assumed to be proportional to the product of the remaining amounts of the chemicals A and B not converted to C. It is observed that 40 grams of chemical C has been formed in 1/1 minutes. How many grams of C will have been formed in 2/1 minutes? 2. For which values of the parameters α and β has the linear system ( ) ( ) dx/dt x + y = dy/dt αx + βy only periodic solutions of period π? 3. Find to the differential equation x 2 y +3xy +y = 8x the solution whose graph at the point with the coordinates (1, 2) has the tangent line x + y = Solve the initial-value problem y(0) = 1, y (t) + 3y(t) = δ(t 2) + { 0, 0 t <, 4, t, where δ is the Dirac delta function (in fact the Dirac distribution).. Sketch a representative selection of curves of the phase portrait of the linear system ( ) ( ) dx/dt x + 4y =. dy/dt x + y 6. An ideal string of normal length can vibrate in a xu-plane. The string is in its both ends (the positions 0 and ) held fixed at displacement level 0. At time 0, the string is in motion such that no point of the string is displaced but has the velocity 2. Find the displacement u of the string for 0 < x <, t > 0, where it is assumed that the displacement obeys the wave equation 9u xx = u tt. 7. Solve the differential equation (x 2 + 1)y + xy = x with y(1) = 2. Also, determine the interval of existence and uniqueness for the solution. 8. Find the number sequence {y n } n=0 which for n 2 satisfies the difference equation 2y n 10y n 1 + y n 2 = 0, and which has the initial values y 0 = 1, y 1 = 2/. Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.

2 MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA134 Differentialekvationer och transformmetoder Datum: Skrivtid: timmar Hjälpmedel: Skrivdon Denna tentamen består av åtta stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt poäng. För godkänd-betygen 3, 4 och krävs erhållna poängsummor om minst 18, 26 respektive 34 poäng. För ECTS-betygen E, D, C, B och A krävs 18, 20, 26, 33 respektive 38. Lösningar förutsätts innefatta ordentliga motiveringar och tydliga svar. Samtliga lösningsblad skall vid inlämning vara sorterade i den ordning som uppgifterna är givna i. 1. Vid tidpunkten 0 finns det 80 gram av kemikalie A och 60 gram av kemikalie B. Kemikalierna blandas varvid de reagerar med varandra i proportionerna 2 : 3 och bildar kemikalie C, dvs för varje gram av slutprodukten C så går det åt 2 gram av A och 3 gram av B. Det antages att energiåtgången vid reaktionen är försumbar i förhållande till de energier som massorna representerar. Reaktionshastigheten antages vara proportionell mot produkten av återstoderna av ämnena A och B. Det observeras att 40 gram av ämne C har bildats efter 1/1 minuter. Hur många gram C kommer att ha bildats efter 2/1 minuter? 2. För vilka värden på parametrarna α och β har det linjära systemet ( ) ( ) dx/dt x + y = dy/dt αx + βy enbart periodiska lösningar med perioden π? 3. Bestäm till differentialekvationen x 2 y + 3xy + y = 8x den lösning vars graf i punkten med koordinaterna (1, 2) har tangenten x + y = Lös begynnelsevärdesproblemet y(0) = 1, y (t) + 3y(t) = δ(t 2) + { 0, 0 t <, 4, t, där δ är Diracs deltafunktion (eg. Dirac-distributionen).. Skissa ett representativt urval av kurvor i fasporträttet till det linjära systemet ( ) ( ) dx/dt x + 4y =. dy/dt x + y 6. En ideal sträng med normallängden kan svänga i ett xu-plan. Strängen är i sina bägge ändar (lägena 0 och ) fastsatt på utslagsnivån 0. Vid tidpunkten 0 befinner sig strängen i rörelse på så sätt att varje punkt på strängen har utslaget 0 och hastigheten 2. Bestäm strängens utslag u för 0 < x <, t > 0, då det antages att utslaget lyder vågekvationen 9u xx = u tt. 7. Lös differentialekvationen (x 2 + 1)y + xy = x med y(1) = 2. Bestäm även existens- och entydighetsintervallet för lösningen. 8. Bestäm den talföljd {y n } n=0 som för n 2 satisfierar differensekvationen 2y n 10y n 1 + y n 2 = 0, och som har begynnelsevärdena y 0 = 1, y 1 = 2/. If you prefer the problems formulated in English, please turn the page.

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7 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN grams ( 4 1t 3) ( 4 3) 2 x C ( t) = 100, where [t] = min t 2. α = β = 1 EXAMINATION IN MATHEMATICS MAA134 Differential Eq:s and Transform Meth. EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2014/1 Maximum points for subparts of the problems in the final examination 1p: Correctly determined the relations between x A, x B, x C 2p: Correctly formulated and correctly solved the DE 1p: Correctly determined the integration constant and the proportional factor of the DE 1p: Correctly determined the value of x 2 ) C ( 1 1p: Correctly noted the solutions of the system are periodic iff the trace τ = 1 + β and the determinant = β α of the system matrix are equal to zero and greater than zero respectively 2p: Correctly noted that the angular frequency ω for periodic solutions is equal to the square root of the determinant of the coefficient matrix 1p: Correctly noted that the period is equal to 2 π ω 1p: Correctly solved the system of equations for the parameters α and β ln( x) y = 2x x y( t) = e 3t + + e 4 3 3( t 2) (1 e U ( t 2) 3( t ) ) U ( t ) One scenario for the first three points p: Correctly determined the differential equation for ~ y, where ~ y ( u) = y( x ) and e u = x 1p: Correctly determined the complementary solution of the differential equation for ~ y 1p: Correctly determined the a particular solution of the differential equation for ~ y Another scenario for the first three points β 1p: Correctly, by testing solutions of the type y = x determined a solution of the homogeneous differential 2 equation x y + 3xy + y = 0 1p: Correctly by reduction of order determined the general solution of homogeneous differential equation 1p: Correctly by variation of parameter determined the general solution of the nonhomogeneous differential equation A scenario for the final two points p: Correctly (in x ) compiled the general solution of the differential equation, and correctly interpreted the condition that the graph includes the point that has the coordinates ( 1,2) 1p: Correctly handled the tangential condition at the given point 1p: Correctly Laplace transformed the differential equation 2p: Correctly prepared for an inverse transformation 2p: Correctly determined a solution of the differential equation 1 (2)

8 . X e 1 t ( t) = c1 e + c2 3t 1p: Correctly determined one of the eigenvalues and the corresponding eigenvectors 1p: Correctly determined the other of the two eigenvalues and the corresponding eigenvectors 1p: Correctly compiled the general solution of the DES 2p: Correctly sketched a representative selection of curves of the phase portrait (, ) 20 n ( 1 ( 1) ) nπ x 3n u x t = 2 sin( )sin( n= 1 = p= 1 3( nπ ) (2 p 1) π (2 p 3(2 sin( 1) π x p )sin( for 0 < x <, t > 0 x + 1 y = x I E = (, ) π t 1) π t ) ) 2p: Correctly solved the x-part of the differential equation 1p: Correctly solved the t-part of the differential equation 2p: Correctly determined the Fourier coefficients, and correctly compiled the solution 2p: Correctly identified the DE as a nonhomogeneous linear equation of order one, correctly written the DE in the standard form, and correctly determined an integrating factor 1p: Correctly solved the DE 1p: Correctly adapted the solution to the initial value 1p: Correctly determined the interval of existence 8. The number sequence terms 1 ( ) n yn = ( n +1) { n} n=0 y has the 1p: Correctly z-transformed the difference equation 1p: Correctly determined the z-transform of the number sequence 2p: Correctly prepared for the inverse transformation 1p: Correctly determined the solution of the difference eq. 2 (2)