2. Find an equation for and sketch the curve which begins at the point P : (3, 1) and which otherwise is given by the linear system 1 = 2

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1 MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Eaminer: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA36 Differential Equations, foundation course Date: Write time: 5 hours Aid: Writing materials, ruler This eamination consists of eight randomly ordered problems each of which is worth at maimum 5 points. The pass-marks 3, 4 and 5 require a minimum of 8, 6 and 34 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 8, 0, 6, 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.. Find, in terms of a power series in, the general solution of the differential equation y + 6 y = 0 in a neighbourhood of 0. Specify eplicitly the terms up to order 9 of the series solution.. Find an equation for and sketch the curve which begins at the point P : (3, ) and which otherwise is given by the linear system ( ) ( d/dt = y ). dy/dt In sketching, the following approimations might be of value to know: e / 0.6, e 0.37, e 3/ 0., e 0.4, e 5/ 0.08, e Solve the differential equation (y 3)y = (y ) with y() = and y () = 3, and specify the interval of eistence of the solution. 4. The differential equation (y ) d + ( y) dy = 0 has an integrating factor which only depends on. Find an equation on the form y = f() for the solution curve that includes the point with the coordinates (e, 0). 5. Find, for > 0, the general solution of the differential equation y y 0y = Classify, for all β 3, 0, the stationary point (origin) of the system ( ) ( ) d/dt 3 + β = y. dy/dt + βy 7. Temperature changes in a thermometer are assumed to be described by Newton s cooling/warming law. A feverish person has with the help of such a thermometer measured its own body temperature to be 39 o C. What does the thermometer read two minutes after the fever measurement if it shows 33 o C one minute after the fever measurement and if the room temperature is o C? It is considered reasonable to suppose that the thermometer s small etent and the corresponding negligible heat content not substantially disturb the temperature of the surrounding medium (the person s body and the air respectively). 8. Solve the initial-value problem y(0) =, y (0) = 3, y (t) 8y (t) + 6y(t) = δ(t 7), where δ is the Dirac delta function (distribution). 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 Eaminator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA36 Differentialekvationer, grundkurs Datum: Skrivtid: 5 timmar Hjälpmedel: Skrivdon, linjal Denna tentamen består av åtta stycken om varannat slumpmässigt ordnade uppgifter som vardera kan ge maimalt 5 poäng. För godkänd-betygen 3, 4 och 5 krävs erhållna poängsummor om minst 8, 6 respektive 34 poäng. För ECTS-betygen E, D, C, B och A krävs 8, 0, 6, 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.. Bestäm, uttryckt i en potensserie i, den allmänna lösningen till differentialekvationen y + 6 y = 0 i en omgivning till 0. Specificera eplicit i serielösningen termerna upp till och med ordning 9.. Bestäm en ekvation för och skissa den kurva som börjar i punkten P : (3, ) och i övrigt ges av det linjära systemet ( ) ( d/dt = y ). dy/dt I skissandet kan följande approimationer tänkas vara av värde att känna till: e / 0.6, e 0.37, e 3/ 0., e 0.4, e 5/ 0.08, e Lös differentialekvationen (y 3)y = (y ) med y() = och y () = 3, och specificera eistensintervallet för lösningen. 4. Differentialekvationen (y ) d+( y) dy = 0 har en integrerande faktor som bara beror av. Bestäm en ekvation på formen y = f() för den lösningskurva som inkluderar punkten med koordinaterna (e, 0). 5. Bestäm, för > 0, den allmänna lösningen till differentialekvationen y y 0y = Klassificera, för alla β 3, 0, den stationära punkten (origo) till systemet ( ) ( ) d/dt 3 + β = y. dy/dt + βy 7. Temperaturförändringar hos en termometer antas kunna beskrivas med Newtons avkylnings- och uppvärmningslag. En febersjuk person har med hjälp av en sådan termometer mätt sin egen kroppstemperatur till att vara 39 o C. Vad visar termometern två minuter efter febermätningen om den visar 33 o C en minut efter febermätningen och om rumstemperaturen är o C? Det kan anses rimligt att antaga att termometerns ringa konstitution och motsvarande försumbara värmeinnehåll ej nämnvärt rubbar temperaturen i det omgivande mediet (personens kropp respektive luften). 8. Lös begynnelsevärdesproblemet y(0) =, y (0) = 3, y (t) 8y (t) + 6y(t) = δ(t 7), där δ är är Diracs deltafunktion (distribution). 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 Eaminer: Lars-Göran Larsson Eamination TEN EXAMINATION IN MATHEMATICS MAA36 Differential Equations, foundation course EVALUATION PRINCIPLES with POINT RANGES Academic Year: 05/6 Maimum points for subparts of the problems in the final eamination.. 3. y = A( + B( ) 9 + ) where A and B are constants 3 t X( t) = e t 9 7 y = 3 I E = ( 3, ) t p: Correctly worked out the derivatives of the power series assumption for the solution, and correctly in both terms of the LHS of the DE shifted the indices of summation so that the series are integrated into one p: Correctly identified the iteration relation for the coefficients of the power series assumption together with the special conditions for the coefficients number and 3 p: Correctly found the values of coefficients number 4 7 in terms of the coefficients number 0 and p: Correctly found the values of coefficients number 8 9 in terms of the coefficients number 0 and p: Correctly stated the general solution of the DE including eplicit terms up to at least order One scenario p: Correctly found the eigenvalue and a corresponding eigenvector p: Correctly found an additional vector p: Correctly compiled the general solution p: Correctly sketched the solution curve Another scenario p: Correctly Laplace transformed the system of DE:s p: Correctly prepared for an inverse transformation by solving for X( s) p: Correctly inverse transformed X( s) p: Correctly sketched the solution curve p: Correctly worked out the substitution y ( ) = u( y( )) and correctly found that the DE can be splitted into two separate DE:s p: Correctly, from the initial values, argued that the case y ( ) = 0 has to be disregarded p: Correctly found the solution of the IVP p: Correctly specified the interval of eistence 4. y = + ( ln( )) p: Correctly found an integrating factor for the DE p: Correctly identified the equations for a potential function φ by which the DE can be reformulated as d φ = 0 p: Correctly found a potential function φ p: Correctly integrated the eact IVP p: Correctly, on the form y = f (), found the solution of the IVP ()

8 5. 5 y = C + C One scenario for the first three points p: Correctly reformulated the DE into a DE for ~ ~ y, where y ( u) = y( ) and e u = p: Correctly found the complementary solution of the DE for ~ y p: Correctly found a particular solution of the DE for ~ y p: Correctly compiled the general solution of the DE for ~ y p: Correctly resubstituted instead of u in the equation for the solution of the DE Another scenario for the first three points β p: Correctly, by testing solutions of the type y = found two solutions of the corresponding homogeneous DE y y 0y = 0 p: Correctly compiled the two solutions into the general solution of y y 0y = 0 p: Correctly by variation of parameter found the general solution of the DE y y 0y = 3 6. β < 3: stable spiral point 3 < β < 0 : saddle point 0 < β < : unstable node β = : degenererate unstable node β > : unstable spiral point p: Correctly classified the SP for β < 3 p: Correctly classified the SP for 3 < β < 0 p: Correctly classified the SP for 0 < β < p: Correctly classified the SP for β = p: Correctly classified the SP for β > 7. 9 o C p: Correctly formulated a DE for the temperture T (counted in o C) of the thermometer at time t (counted in minutes) p: Correctly solved the DE p: Correctly adapted the solution of the DE to the given conditions p: Correctly found the temperature at time minutes 4t 4( t 7) 8. y( t) = ( 7t) e + ( t 7) e U ( t 7) p: Correctly Laplace transformed the differential equation p: Correctly prepared for an inverse transformation p: Correctly inverse transformed the not time-delayed part p: Correctly inverse transformed the time-delayed part ()