M ¨ALARDALEN UNIVERSITY
School of Education, Culture and Communication Department of Applied Mathematics
Examiner : Lars-G¨oran Larsson
EXAMINATION IN MATHEMATICS MAA316 Differential Equations, foundation course Date: 2018-03-21 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 the general solution of the linear system
dx/dt dy/dt dz/dt
=
2x + 3y 3x − z 6y + 2z
.
2. Find to the differential equation 2y00+ (y0)2 = 0 the solution whose graph at the point with the coordinates (1, 0) has the tangent line x + y = 1 .
3. Find to the differential equation y0 = 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)y00− 3xy0+ y = 0, y(0) = −3, y0(0) = 2
in a neighbourhood of 0. In the series solution, specify explicitly the terms up to at least degree 5.
5. Find all stationary points of the system
dx/dt dy/dt
=
y2− xy 6 + x − y2
,
and classify each of them as unstable, stable or asymptotically stable.
6. Find, for x > 0, the general solution of the differential equation xy00+ (2x − 3)y0+ (x − 3)y = 0 .
7. Solve the initial-value problem
y(0) = 1 , y0(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 90oC are hanged airy for cooling in a well-ventilated room where the air tempera- ture through various techniques is kept constant at 10oC. Packaging of a thin breads takes place when they have cooled to 15oC. 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 30oC? 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 f¨oredrar uppgifterna formulerade p˚a svenska, var god v¨and p˚a bladet.
M ¨ALARDALENS H ¨OGSKOLA
Akademin f¨or utbildning, kultur och kommunikation Avdelningen f¨or till¨ampad matematik
Examinator : Lars-G¨oran Larsson
TENTAMEN I MATEMATIK MAA316 Differentialekvationer, grundkurs Datum: 2018-03-21 Skrivtid: 5 timmar Hj¨alpmedel: Skrivdon, linjal
Denna tentamen best˚ar av ˚atta stycken om varannat slumpm¨assigt ordnade uppgifter som vardera kan ge maximalt 5 po¨ang.
F¨or godk¨and-betygen 3, 4 och 5 kr¨avs erh˚allna po¨angsummor om minst 18, 26 respektive 34 po¨ang. F¨or ECTS-betygen E, D, C, B och A kr¨avs 18, 20, 26, 33 respektive 38. L¨osningar f¨oruts¨atts innefatta ordentliga motiveringar och tydliga svar. Samtliga l¨osningsblad skall vid inl¨amning vara sorterade i den ordning som uppgifterna ¨ar givna i.
1. Best¨am den allm¨anna l¨osningen till det linj¨ara systemet
dx/dt dy/dt dz/dt
=
2x + 3y 3x − z 6y + 2z
.
2. Best¨am till differentialekvationen 2y00+ (y0)2 = 0 den l¨osning vars graf i punkten med koordinaterna (1, 0) har tangenten x + y = 1 .
3. Best¨am till differentialekvationen y0 = x/y + 2y/x den l¨osning som satisfierar villkoret y(1) = −2. Identifiera och ange ¨aven existensintervallet f¨or l¨osningen.
4. Best¨am, uttryckt som en potensserie i x, l¨osningen till begynnelsev¨ardesproblemet (x + 2)y00− 3xy0+ y = 0, y(0) = −3, y0(0) = 2
i en omgivning till 0. Specificera explicit i seriel¨osningen termerna upp till och med ˚atminstone grad 5.
5. Best¨am alla station¨ara punkter till systemet
dx/dt dy/dt
=
y2− xy 6 + x − y2
,
och klassificera var och en av dem som instabil, stabil eller asymptotiskt stabil.
6. Best¨am, f¨or x > 0, den allm¨anna l¨osningen till differentialekvationen xy00+ (2x − 3)y0+ (x − 3)y = 0 .
7. L¨os begynnelsev¨ardesproblemet
y(0) = 1 , y0(t) + 3y(t) = δ(t − 2) +
0 , 0 ≤ t < 4 , 1 , 4 ≤ t < 7 , 0 , t ≥ 7 .
d¨ar δ ¨ar Diracs ”deltafunktion” (eg. Dirac-distributionen).
8. Nybakade tunnbr¨od som direkt efter gr¨addningen har temperaturen 90oC h¨angs upp luftigt f¨or avsvalning i ett v¨alventilerat rum d¨ar lufttemperaturen genom diverse teknik h˚alls konstant vid 10oC. Inpaketering av tunnbr¨od sker n¨ar de har svalnat till 15oC. Hur l˚ang tid tar det fr˚an det att ett tunnbr¨od har f¨ardiggr¨addats till dess att det kan paketeras om temperaturen hos ett avsvalnande tunnbr¨od an- tas kunna beskrivas med Newtons avkylnings- och uppv¨armningslag, och om br¨od efter en minut i avsvalningsrummet erfarenhetsm¨assigt har temperaturen 30oC?
Det kan anses rimligt att antaga att br¨odens relativt f¨orsumbara v¨armeinneh˚all ej n¨amnv¨art rubbar lufttemperaturen i avsvalningsrummet.
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 IN MATHEMATICS
MAA316 Differential Equations, foundation course EVALUATION PRINCIPLES with POINT RANGES Academic Year: 2017/18
Examination TEN1 – 2018-03-21 Maximum points for subparts of the problems in the final examination
1. 𝑋𝑋(𝑡𝑡)
= 𝑐𝑐
1�
−112
� 𝑒𝑒
−𝑡𝑡+ 𝑐𝑐
2�
1 0 3
� 𝑒𝑒
2𝑡𝑡+ 𝑐𝑐
3�
3 1 6
� 𝑒𝑒
3𝑡𝑡1p: 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−𝑥𝑥2� 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= �
√51, ∞�
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 y
−1( 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 +
34𝑥𝑥
2+
245𝑥𝑥
3+
485𝑥𝑥
4+
961𝑥𝑥
5+ ⋯
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)
6. 𝑦𝑦 = 𝐶𝐶
1𝑒𝑒
−𝑥𝑥+ 𝐶𝐶
2𝑥𝑥
4𝑒𝑒
−𝑥𝑥1p: 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
7. 𝑦𝑦 = 𝑒𝑒
−3𝑡𝑡+ 𝑒𝑒
−3(𝑡𝑡−2)𝑈𝑈(𝑡𝑡 − 2)
+
13�1 − 𝑒𝑒
−3(𝑡𝑡−4)�𝑈𝑈(𝑡𝑡 − 4)
−
13�1 − 𝑒𝑒
−3(𝑡𝑡−7)�𝑈𝑈(𝑡𝑡 − 7)
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
oC) 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
oC after 2 minutes of cooling
2 (2)
