1. Find, for x > 0, the general solution of the differential equation. dy/dt 4xy + 10y + 6y 2,

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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. Find, for x > 0, the general solution of the differential equation xy (4x + 1)y + 2(2x + 1)y = Solve the integral equation y(t) = 5e 4t Find all stationary points of the system ( ) ( ) dx/dt 2x x = 2 + 3xy dy/dt 4xy + 10y + 6y 2, t 0 (ξ t)y(ξ) dξ on the interval [0, ). and classify each of them as unstable, stable or asymptotically stable. 4. At time 0, there are 20 grams of chemical A and 50 grams of chemical B. The chemicals are combined whereby they react together in the proportions 1 : 2 forming the chemical C. It is assumed that the energy consumed in the reaction is negligible in relation to the energies which the masses represent, i.e. it is assumed that for each 3 grams of the final product C, 1 gram of A and 2 grams of B will be used. 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 25 grams of chemical C has been formed in 1 minute. How many grams of C will have been formed in 2 minutes? 5. Find the (unique) solution of the initial-value problem y 3y + 2y = cos(e x ), y(ln( 1 π )) = y (ln( 1 π )) = 1 π. 6. Find an equation for and sketch the curve that starts at the point P : (3, 1) and that satisfies the linear system ( ) ( ) dx/dt 3x 6y =. dy/dt 3x 3y Especially, state the direction in which the curve is passed through for increasing values of t, and classify the stationary point (the origin) of the system. 7. Find to the differential equation x dy + 2y = (xy)2 the solution that satisfies dx the condition y(e) = 1/e 2. Also, find the interval of existence for the solution. 8. Classify all singular points of the differential equation x 3 (x 2 9) 2 y + 2x 2 (x 3)y + 5(x + 3)y = 0. 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. Bestäm, för x > 0, den allmänna lösningen till differentialekvationen xy (4x + 1)y + 2(2x + 1)y = Lös integralekvationen y(t) = 5e 4t Bestäm alla stationära punkter till systemet ( ) ( ) dx/dt 2x x = 2 + 3xy dy/dt 4xy + 10y + 6y 2, t 0 (ξ t)y(ξ) dξ på intervallet [0, ). och klassificera var och en av dem som instabil, stabil eller asymptotiskt stabil. 4. Vid tidpunkten 0 finns det 20 gram av kemikalie A och 50 gram av kemikalie B. Kemikalierna blandas varvid de reagerar med varandra i proportionerna 1 : 2 och bildar kemikalie C. Det antages att energiåtgången vid reaktionen är försumbar i förhållande till de energier som massorna representerar, dvs det antages att för varje 3 gram av slutprodukten C så går det åt 1 gram av A och 2 gram av B. Reaktionshastigheten antages vara proportionell mot produkten av återstoderna av ämnena A och B. Det observeras att 25 gram av ämne C har bildats efter 1 minut. Hur många gram C kommer att ha bildats efter 2 minuter? 5. Bestäm (den entydiga) lösningen till begynnelsevärdesproblemet y 3y + 2y = cos(e x ), y(ln( 1 π )) = y (ln( 1 π )) = 1 π. 6. Bestäm en ekvation för och skissa den kurva som börjar i punkten P : (3, 1) och som satisfierar det linjära systemet ( ) ( ) dx/dt 3x 6y =. dy/dt 3x 3y Ange speciellt i vilken riktning kurvan genomlöpes för ökande värden på t, och klassificera den stationära punkten (origo) till systemet. 7. Bestäm till differentialekvationen x dy + 2y = (xy)2 den lösning som satisfierar dx villkoret y(e) = 1/e 2. Bestäm även existensintervallet för lösningen. 8. Klassificera alla singulära punkter till differentialekvationen x 3 (x 2 9) 2 y + 2x 2 (x 3)y + 5(x + 3)y = 0. 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: 2016/17 Maximum points for subparts of the problems in the final examination 1. y y = e of the DE 2p: Correctly performed a reduction of order in the DE, and correctly solved the reduced DE 2p: Correctly compiled the general solution of the DE 2x 2 2x = C1 e + C2x e 1p: Correctly found e.g. the solution 2x 4t 2. y( t) = 4e + cos(2t) 2sin(2t) 1p: Correctly interpreted the integral in the right-hand side of the integral equation as a certain convolution 1p: Correctly Laplace transformed the integral equation 1p: Correctly prepared for an inverse transformation 2p: Correctly found the solution of the integral equation 3. P 1 : (0, 0) is an unstable SP 2p: Correctly found the stationary points of the nonlinear 5 P 2 : (0, ) is an asymptotically stable SP system of differential equations 3 1p: Correctly classified two of the stationary points P 3 : (2,0) is an unstable SP 1p: Correctly classified one more of the stationary points 1 P 4 : (3, ) is an asymptotically stable SP 1p: Correctly classified the last of the stationary points grams C 8 t ( ) ( 8 t ) 7 5 x ( t) = 60, where [t] = min 1p: Correctly specified 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 C (2 ) x 5. 2 x y = e e (1 + cos( e x )) 1p: Correctly found the 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 adapted the general solution to the initial values 3cos(3t) + sin(3t) 6. X( t ) = cos(3t) + 2sin(3t) 1p: Correctly determined the eigenvalues and the eigenvectors of the coefficient matrix 2p: Correctly compiled the general solution of the DES, and correctly adapted to the initial condition 1p: Correctly, with direction, sketched the solution curve 1p: Correctly classified the stationary point of the system The stationary point is a center 1 (2)

8 7. y = x 2 I E = (0, e 1 (2 ln( x)) 2 ) 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 8. 3 is a regular singular point of the DE 0 and 3 are irregular singular points of the DE 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 2 (2)