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: 016-06-0 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, 6 and 34 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 18, 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. 1. Solve, for t 0, the integral equation t 0, 0 t <, y(t) + 9 (ξ t) y(ξ) dξ = 1, t < 5, 0 0, t 5.. At time 0, there are 100 grams of a substance which decays with a rate proportional partly to the cube of the remaining amount of substance (in grams counted) at time t (in minutes counted) and partly to an environmental factor e t. At time ln(5/4) minutes there remains 50 grams of the substance. How many grams of the substance will never decay? 3. Find, in terms of a power series in x, the solution of the initial-value problem y + (x 1)y + y = 0, y(0) = 0, y (0) = 1 in a neighbourhood of 0. In the series solution, specify explicitly the terms up to at least order 4. 4. Find all stationary points of the system ( ) ( ) dx/dt 6x 3x = 4xy dy/dt 3xy y and classify each of them with respect to the corresponding linearized system. 5. Find to the differential equation x dy + xy = y dx the solution whose graph contains the point with the coordinates (1, 3 ). Also, determine the interval of existence of the solution. 6. Find to the differential equation y + y = 1 cos 3 (x) the solution whose graph at the point with the coordinates (0, 1 ) has the tangent line x + 1 = y. 7. Find, for x > 3, the general solution of the differential equation (x + 3)y + (x + )y y = 0. 8. Find the general solution of the linear system ( ) dx/dt = dy/dt ( x + y 4x + y and sketch the two solution curves γ A and γ B obtained with the initial values (x(0), y(0)) = (, ) and (x(0), y(0)) = (, 1) respectively. ), Om du föredrar uppgifterna formulerade på svenska, var god vänd på bladet.
MÄLARDALENS HÖGSKOLA Akademin för utbildning, kultur och kommunikation Avdelningen för tillämpad matematik Examinator: Lars-Göran Larsson TENTAMEN I MATEMATIK MAA316 Differentialekvationer, grundkurs Datum: 016-06-0 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, 6 respektive 34 poäng. För ECTS-betygen E, D, C, B och A krävs 18, 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. 1. Lös, för t 0, integralekvationen t 0, 0 t <, y(t) + 9 (ξ t) y(ξ) dξ = 1, t < 5, 0 0, t 5.. Vid tidpunkten 0 finns det 100 gram av ett ämne som sönderfaller i en takt som är proportionell mot dels kuben av återstoden av ämnet (i gram räknad) vid tidpunkten t (i minuter räknad) och dels en miljöfaktor e t. Vid tiden ln(5/4) minuter återstår 50 gram av ämnet. Hur många gram av ämnet kommer aldrig att sönderfalla? 3. Bestäm, uttryckt som en potensserie i x, lösningen till begynnelsevärdesproblemet y + (x 1)y + y = 0, y(0) = 0, y (0) = 1 i en omgivning till 0. Specificera explicit i serielösningen termerna upp till och med åtminstone ordning 4. 4. Bestäm alla stationära punkter till systemet ( ) ( ) dx/dt 6x 3x = 4xy dy/dt 3xy y och klassificera var och en av dem med avseende på motsvarande linjariserade system. 5. Bestäm till differentialekvationen x dy + xy = y dx den lösning vars graf innehåller punkten med koordinaterna (1, 3 ). Bestäm även existensintervallet för lösningen. 6. Bestäm till differentialekvationen y + y = 1 cos 3 (x) den lösning vars graf i punkten med koordinaterna (0, 1 ) har tangenten x+1 = y. 7. Bestäm, för x > 3, den allmänna lösningen till differentialekvationen (x + 3)y + (x + )y y = 0. 8. Bestäm den allmänna lösningen till det linjära systemet ( ) ( ) dx/dt x + y =, dy/dt 4x + y och skissa de två lösningskurvor γ A och γ B som fås med begynnelsevillkoren (x(0), y(0)) = (, ) respektive (x(0), y(0)) = (, 1). If you prefer the problems formulated in English, please turn the page.
MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson Examination TEN1 016-06-0 EXAMINATION IN MATHEMATICS MAA316 Differential Equations, foundation course EVALUATION PRINCIPLES with POINT RANGES Academic Year: 015/16 Maximum points for subparts of the problems in the final examination 1. y( t) cosh(3( t )) U( t ) cosh(3( t 5)) U( t 5) 1p: Correctly interpreted the integral in the left-hand side of the differential equation as a certain convolution 1p: Correctly Laplace transformed the differential equation 1p: Correctly prepared for an inverse transformation p: Correctly found the solution of the differential equation. 5 grams 100 x( t), where [t] min t 16 15e 1p: Correctly formulated an equation for the amount of substance x at time t 1p: Correctly solved the DE 1p: Correctly adapted the solution of the DE to the given conditions p: Correctly found the amount of substance which never decay 1 3 1 4 3. y x 1 x 6 x x 6 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 p: Correctly found the terms up to at least order 4 4. P 1 : (0,0) is a saddle point P : (,0) is a saddle point P (,1) is a stable spiral point 3 : 3 of the corresponding linearized system of differential equations. p: Correctly found the stationary points of the system of differential equations 3p: Correctly classified the stationary points with respect of the corresponding linearized systems of differential equations (1p for each correctly classified stationary point) 5. 3x y x 3 1 I E ( 1, ) 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 p: Correctly solved the DE 1p: Correctly found the solution of the IVP 1p: Correctly found the interval of existence 1 1 6. y sin( x) cos( x) 1p: Correctly found the solution of the associated homogeneous equation 3p: Correctly, by variation of parameters, found the general solution of the nonhomogeneous equation 1p: Correctly adapted the general solution to the (implicitly given) initial values 1 ()
x 7. y C e C x ) ( 1 x 1p: Correctly found e.g. the solution y1 e of the differential equation (DE) p: Correctly performed a reduction of order in the DE, and correctly solved the reduced DE p: Correctly compiled the general solution of the DE 8. X 1 1 1 e 4 3t ( t) c1 e c t 1p: Correctly determined one of the eigenvalues and the corresponding eigenspace 1p: Correctly determined the other of the two eigenvalues and the corresponding eigenspace 1p: Correctly compiled the general solution of the DES 1p: Correctly sketched the curve A 1p: Correctly sketched the curve B ()