1. For which values of the parameters α and β has the linear system. periodic solutions? Which are the values if the period equals 4π?

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1 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA316 Differential Equations, foundation course Date: Write time: 5 hours Aid: Writing materials, ruler This examination consists of eight randomly ordered problems each of which is worth at maximum 5 points. The pass-marks 3, 4 and 5 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. For which values of the parameters α and β has the linear system ( ) ( ) dx/dt 3x + αy = dy/dt x + βy periodic solutions? Which are the values if the period equals 4π? 2. Classify all singular points of the differential equation (x 2 4) 2 (x 2 + 2x)y + (x 2 4)(x + 2)y + xy = At time 0, there are 10 grams of a substance which decays with a rate proportional to the product of partly the square of the remaining amount of substance (in grams counted) at time t (in minutes counted) and partly the time-dependent factor 1/ 4 + t. After five minutes, 9 grams remain of the substance. How many grams of the substance remains after an hour? 4. Find to the differential equation (2x + 1)y + 4xy 4y = 0, x > 1 2 the solution which at the point with the coordinates (0, 1) has the tangent line y = 4x Solve, for t 0, the integral equation 3y(t) = 5 t 0 sin(3ξ) y(t ξ) dξ + { 0, 0 t < 5, 4, t Sketch a representative selection of curves of the phase portrait of the linear system ( ) ( ) dx/dt 6x + 2y =. dy/dt 8x + 2y 7. Find the general solution of differential equation y 6y + 9y = 6e3x x Find to the differential equation y + y(1 + e x y) = 0 the solution which satisfies the condition y(0) = 1. Also, find the interval of existence for the solution. 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 MAA316 Differentialekvationer, grundkurs Datum: Skrivtid: 5 timmar Hjälpmedel: Skrivdon, linjal Denna tentamen består av åtta stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maximalt 5 poäng. För godkänd-betygen 3, 4 och 5 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. För vilka värden på parametrarna α och β har det linjära systemet ( ) ( ) dx/dt 3x + αy = dy/dt x + βy periodiska lösningar? Vilka är värdena om perioden är lika med 4π? 2. Klassificera alla singulära punkter till differentialekvationen (x 2 4) 2 (x 2 + 2x)y + (x 2 4)(x + 2)y + xy = Vid tidpunkten 0 finns det 10 gram av ett ämne som sönderfaller i en takt som är proportionell mot produkten av dels kvadraten av återstoden av ämnet (i gram räknad) vid tidpunkten t (i minuter räknad) och dels den tidsberoende faktorn 1/ 4 + t. Efter fem minuter återstår 9 gram av ämnet. Hur många gram av ämnet återstår efter en timme? 4. Bestäm till differentialekvationen (2x + 1)y + 4xy 4y = 0, x > 1 2 den lösning som i punkten med koordinaterna (0, 1) har tangenten y = 4x Lös, för t 0, integralekvationen 3y(t) = 5 t 0 sin(3ξ) y(t ξ) dξ + { 0, 0 t < 5, 4, t Skissa ett representativt urval av kurvor i fasporträttet till det linjära systemet ( ) ( ) dx/dt 6x + 2y =. dy/dt 8x + 2y 7. Bestäm den allmänna lösningen till differentialekvationen y 6y + 9y = 6e3x x Bestäm till differentialekvationen y + y(1 + e x y) = 0 den lösning som satisfierar villkoret y(0) = 1. Bestäm även existensintervallet för lösningen. 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 The linear system has periodic solutions iff α > 9 and β = 3. If α = then the period equals 4 π 4 EXAMINATION IN MATHEMATICS MAA316 Differential Equations, foundation course EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2016/17 Maximum points for subparts of the problems in the final examination 1p: Correctly stated that the system has periodic solutions iff > 0 and τ = 0, where and τ are the determinant and the trace respectively of the system matrix 1p: Correctly, in terms of α and β, formulated the conditions for having periodic solutions 1p: Correctly noted that the period is equal to 2 π ω, where ω is the angular frequency 1p: Correctly noted that the angular frequency ω is equal to 1p: Correctly found the value of the parameter α for which the system has solutions with the period equal to 4 π 2. 2 is a regular singular point of the DE. 0 and 2 are irregular singular points of the DE 3. 6 grams remains after an hour 9 x( t) = 10, where [t] = min t 1p: Correctly found the singular points of the linear DE 1p: Correctly classified one of the singular points 2p: Correctly classified one more of the singular points 1p: Correctly classified the last of the singular points 1p: Correctly formulated an equation for the amount of substance x at time t 1p: Correctly solved the DE 2p: Correctly adapted the solution of the DE to the given conditions 1p: Correctly found the amount of substance that remains after an hour 4. y 2x = 2x e 1p: Correctly found one solution of the DE 2p: SCENARIO 1: Correctly found one more solution of the DE such that the two solutions constitute a linear independent set of solutions, OR SCENARIO 2: correctly performed a reduction of order in the DE, and correctly solved the reduced DE 1p: Correctly compiled the general solution of the DE 1p: Correctly adapted the general solution to the IV:s y 1 3 1p: Correctly Laplace transformed the integral in the RHS of the DE 1p: Correctly Laplace transformed the remaining terms of the DE 1p: Correctly prepared for an inverse transformation 2p: Correctly found the solution of the integral equation 5. ( t) = ( 9 5cos(2( t 5)) ) U ( t 5) 1 (2)

8 6. 1 X( t) = c1 e 2 2t t + c2 2t e 2t 1p: Correctly found the double eigenvalue of the matrix and a corresponding eigenmatrix 1p: Correctly found a supplementary column matrix which together with the chosen eigenmatrix forms a linearly independent set of column matrices 1p: Correctly compiled the general solution of the DES 2p: Correctly sketched a representative selection of curves of the phase portrait 7. y 2 3x = ( C1 + C2x + x ) e 1p: Correctly found the general solution of the associated homogeneous equation 1p: Correctly, by variation of parameters, found the antiderivative expressions for the variable parameters 1p: Correctly found the explict expression for one of the two variable parameters 1p: Correctly found the explict expression for the other of the two variable parameters, and correctly summarized the general solution of the differential equation 1p: Correctly summarized the general solution of the differential equation 8. x e y = x +1 I E = ( 1, ) 1p: Correctly identified the DE as a Bernoulli equation, and correctly worked out a suitable substitution 1 y ( x) = u( x) 2p: Correctly solved the DE 1p: Correctly found the solution of the IVP 1p: Correctly found the interval of existence 2 (2)