2. Find to the differential equation 2y + (y ) 2 = 0 the solution whose graph at the point with the coordinates (1, 0) has the tangent line x + y = 1.

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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. dx/dt 2x + 3y 1. Find the general solution of the linear system dy/dt = 3x z. dz/dt 6y + 2z 2. Find to the differential equation 2y + (y ) 2 = 0 the solution whose graph at the point with the coordinates (1, 0) has the tangent line x + y = Find to the differential equation y = x/y + 2y/x the solution that satisfies the condition y(1) = 2. Also, identify and state the interval of existence of the solution. 4. Find, in terms of power series in x, the solution of the initial-value problem (x + 2)y 3xy + y = 0, y(0) = 3, y (0) = 2 in a neighbourhood of 0. In the series solution, specify explicitly the terms up to at least degree Find all stationary points of the system ( ) ( ) dx/dt y = 2 xy dy/dt 6 + x y 2, and classify each of them as unstable, stable or asymptotically stable. 6. Find, for x > 0, the general solution of the differential equation xy + (2x 3)y + (x 3)y = Solve the initial-value problem y(0) = 1, y (t) + 3y(t) = δ(t 2) + { 0, 0 t < 4, 1, 4 t < 7, 0, t 7. where δ is the Dirac delta function (in fact the Dirac distribution). 8. Freshly baked thin breads which immediately after baking has the temperature 90 o C are hanged airy for cooling in a well-ventilated room where the air temperature through various techniques is kept constant at 10 o C. Packaging of a thin breads takes place when they have cooled to 15 o C. How long does it take from that a thin bread has been baked until it can be packaged if the temperature of a cooling thin bread is assumed to be described by Newton s cooling/warming law, and if breads after one minute in the cooling room experientially have the temperature 30 o C? It can be considered reasonable to assume that the bread s relatively negligible heat content does not significantly rub the air temperature in the cooling room. 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. dx/dt 2x + 3y 1. Bestäm den allmänna lösningen till det linjära systemet dy/dt = 3x z. dz/dt 6y + 2z 2. Bestäm till differentialekvationen 2y + (y ) 2 = 0 den lösning vars graf i punkten med koordinaterna (1, 0) har tangenten x + y = Bestäm till differentialekvationen y = x/y + 2y/x den lösning som satisfierar villkoret y(1) = 2. Identifiera och ange även existensintervallet för lösningen. 4. Bestäm, uttryckt som en potensserie i x, lösningen till begynnelsevärdesproblemet (x + 2)y 3xy + y = 0, y(0) = 3, y (0) = 2 i en omgivning till 0. Specificera explicit i serielösningen termerna upp till och med åtminstone grad Bestäm alla stationära punkter till systemet ( ) ( ) dx/dt y = 2 xy dy/dt 6 + x y 2, och klassificera var och en av dem som instabil, stabil eller asymptotiskt stabil. 6. Bestäm, för x > 0, den allmänna lösningen till differentialekvationen xy + (2x 3)y + (x 3)y = Lös begynnelsevärdesproblemet y(0) = 1, y (t) + 3y(t) = δ(t 2) + { 0, 0 t < 4, 1, 4 t < 7, 0, t 7. där δ är Diracs deltafunktion (eg. Dirac-distributionen). 8. Nybakade tunnbröd som direkt efter gräddningen har temperaturen 90 o C hängs upp luftigt för avsvalning i ett välventilerat rum där lufttemperaturen genom diverse teknik hålls konstant vid 10 o C. Inpaketering av tunnbröd sker när de har svalnat till 15 o C. Hur lång tid tar det från det att ett tunnbröd har färdiggräddats till dess att det kan paketeras om temperaturen hos ett avsvalnande tunnbröd antas kunna beskrivas med Newtons avkylnings- och uppvärmningslag, och om bröd efter en minut i avsvalningsrummet erfarenhetsmässigt har temperaturen 30 o C? Det kan anses rimligt att antaga att brödens relativt försumbara värmeinnehåll ej nämnvärt rubbar lufttemperaturen i avsvalningsrummet. 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 EXAMINATION IN MATHEMATICS MAA316 Differential Equations, foundation course EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2017/18 Maximum points for subparts of the problems in the final examination 1. XX(tt) = cc ee tt + cc ee 2tt + cc ee 3tt 1p: Correctly found one of the eigenvalues and the corresponding eigenspace 1p: Correctly found a second of the eigenvalues and the corresponding eigenspace 1p: Correctly found the third of the eigenvalues and the corresponding eigenspace 2p: Correctly compiled the general solution of the DES 2. yy = 2 ln 3 xx 2 1p: Correctly worked out the substitution yy (xx) = uu(yy(xx)) and correctly found that the DE can be divided into two separate DE:s where yy (xx) = 0 can be disregarded since it contradicts the initial value yy (1) = 1 1p: Correctly solved the remaining (linear) DE for u 1p: Correctly adapted u to the initial value uu(yy(1)) = yy (1) 1p: Correctly solved the (separable) DE for yy 1p: Correctly adapted y to the initial value yy(1) = 0 3. yy = xx 5xx 2 1 II EE = 1 5, 1p: Correctly identified the DE as either a homogeneous equation or a Bernoulli equation, and correctly worked out a suitable substitution, preferable y ( x) = xu( x) or 1 y ( x) = u( x) respectively 2p: Correctly solved the DE 1p: Correctly found the solution of the IVP 1p: Correctly found the interval of existence 4. yy = 3 + 2xx xx xx xx xx5 + 1p: Correctly worked out the derivatives of the power series assumption for the solution, and correctly inserted all terms into the DE 1p: Correctly shifted the indices of summation of the series so that the sum of the series are brought into one series, and correctly identified the iteration relations for the coefficients of the power series of the solution 1p: Correctly adapted to the initial values 2p: Correctly found the terms up to at least order 5 5. PP 1 : ( 6,0) is an unstable SP PP 2 : ( 2, 2) is an unstable SP PP 3 : (3,3) is an asymptotically stable SP 2p: Correctly found the stationary points of the nonlinear system of differential equations 1p: Correctly classified one of the stationary points 1p: Correctly classified one more of the stationary points 1p: Correctly classified the last of the stationary points 1 (2)

8 6. yy = CC 1 ee xx + CC 2 xx 4 ee xx 7. yy = ee 3tt + ee 3(tt 2) UU(tt 2) ee 3(tt 4) UU(tt 4) ee 3(tt 7) UU(tt 7) 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 SCENARIO 2: Correctly performed a reduction of order in the DE, and correctly solved the reduced DE 2p: Correctly compiled the general solution of the DE 1p: Correctly Laplace transformed the differential equation 1p: Correctly prepared for an inverse transformation 1p: Correctly inverse transformed the terms corresponding to the initial value and to the Dirac distribution 2p: Correctly inverse transformed the remaining terms 8. 2 minutes 1p: Correctly formulated a DE for the temperture T (counted in o C) of the thin bread at time t (counted in minutes) of the cooling process 1p: Correctly solved the DE 2p: Correctly adapted the solution of the DE to the given conditions 1p: Correctly found that the thin bread has temperature 15 o C after 2 minutes of cooling 2 (2)