StdNum: 13; DE, Teszt 1, Majus 9.
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−4)(y−3)y, y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 6 B: 7 C: 8 D: 9 E: 10
Subexercise (B). Write down the linear approximation of the DE around the smallestyf ix fixedpoint as dtd∆y= a∆y ! How much isa?
MCQ. A: 12 B: 13 C: 14 D: 15 E: 16
Subexercise (C). With the given initial condition fory(0), how much is limt→∞y(t) ? (If limt→tcy(t) =±∞for sometc>0, then answer±∞.)
MCQ. A: 4 B: ∞ C: 3 D: 0 E:−∞
Subexercise (D). With the given initial condition for y(0), how much is limt→−∞y(t) ? (If limt→tcy(t) =±∞
for sometc<0, then answer±∞.) MCQ. A:−∞ B: ∞C: 3 D: 0 E: 4
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−4)(y−3)y, y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 6 B: 7 C: 8 D: 9 E: 10
Subexercise (B). Write down the linear approximation of the DE around the smallestyf ix fixedpoint as dtd∆y= a∆y ! How much isa?
MCQ. A: 12 B: 13 C: 14 D: 15 E: 16
Subexercise (C). With the given initial condition fory(0), how much is limt→∞y(t) ? (If limt→tcy(t) =±∞for sometc>0, then answer±∞.)
MCQ. A: 4 B: ∞ C: 3 D: 0 E:−∞
Subexercise (D). With the given initial condition for y(0), how much is limt→−∞y(t) ? (If limt→tcy(t) =±∞
for sometc<0, then answer±∞.) MCQ. A:−∞ B: ∞C: 3 D: 0 E: 4
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
-0.2 0.0 0.2 0.4 0.6 0
1 2 3 4
Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y =
f1 f2
=
(y1+ 1) (y2+ 1) y1−1
.
Subexercise (B). Write down the linear approximation of the DE around the~yf ix as dtd∆~y=A∆~y ! How much is the sum of the elements ofA ?
MCQ. A: 2 B: 3 C: 4 D: 5 E: 6
Exercise (4). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−4 0 4 −5
, d
dt~y =A~y, ~y(0) = 4
5
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -9 B: -8 C: -7 D: -6 E: -5
Subexercise (B). Find the~v1 and~v2 eigenvectors corresponding to the λ1< λ2 eigenvalues. Normalize them by the condition (~vi)2 = 1, i= 1,2.Assemble them into a matrixS = (~v1, ~v2) matrixot. How much is the sum of the elements ofS ?
MCQ. A: 14 B: 34 C: 54 D: 74 E: 94
Subexercise (C). The solution of the DE can be written as
~ y(t) =
2
X
i=1
Cieλit~vi.
How much isC1, if~y(0) satisfies the given initial condition?
MCQ. A:−13 B: −11 C: −9 D:−8 E:−7
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.00786023 B: 0.00883018 C: 0.00991982 D: 0.0111439 E: 0.0125191 Exercise (5). Hom.Lin. DE. Overdamped oscillator
Let
¨
y+ay˙+by= 0, A=
0 1
−20 −9
, d
dt~y=A~y, ~y(0) = 4
5
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -9 B: -8 C: -7 D: -6 E: -5
Subexercise (B). Find the~v1 and~v2 eigenvectors corresponding to the λ1< λ2 eigenvalues. Normalize them by the condition (~vi)1 = 1, i= 1,2.Assemble them into a matrixS = (~v1, ~v2) matrixot. How much is the sum of the elements ofS ?
MCQ. A:−12 B: −11 C: −10 D:−9 E: −7
Subexercise (C). The solution of the DE can be written as
~ y(t) =
2
X
i=1
Cieλit~vi.
How much isC1, if~y(0) satisfies the given initial condition?
MCQ. A:−23 B: −21 C: −19 D:−18 E: −17
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: -0.0393011 B: -0.0441509 C: -0.0495991 D: -0.0557196 E: -0.0625954
Exercise (6). Hom.Lin. DE. Jordan decomposition.
Theorem: For any complex square A matrix there exists an S such that SAS−1 is block diagonal with blocks choosen from the following list
J1 = λ
, J2 = λ 1
0 λ
, J3=
λ 1 0
0 λ 1 0 0 λ
, . . . .
Subexercise (B). Constant accerelation.
d3y
dt3 = 0, d dt~y =
0 1 0 0 0 1 0 0 0
~y=A~y.
(Here~y= (y,y,˙ y)¨T.) How much is (exp(1.395A))13 ?
MCQ. A: 0.866132 B: 0.973013 C: 1.09308 D: 1.22797 E: 1.3795 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−1 0 2 −1
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(1.395A))21 ?
MCQ. A: 0.615501 B: 0.691454 C: 0.77678 D: 0.872634 E: 0.980317 Exercise (7). Impulse response, distributions.
Dirac-delta: δ(t) = 0, hat6= 0, Z ∞
−∞
δ(t)dt= 1.
Heaviside theta: θ(t) =
(0, hat <0, 1, hat >0,
????: K(t) =
(0, hat <0, t, hat >0., hf(t), φ(t)i=
Z −∞
∞
f(t)φ(t)dt hf0(t), φ(t)i=
Z −∞
∞
f0(t)φ(t)dt=− Z −∞
∞
f(t)φ0(t)dt=−hf(t), φ0(t)i Then
θ0(t) =δ(t),
K0(t) =θ(t), K00(t) =θ0(t) =δ(t).
Subexercise (C).
G0(t) + 3G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.322156 B: 0.36191 C: 0.40657 D: 0.45674 E: 0.513102 Subexercise (E).
G00(t) + 2G(t) =δ(t), G(−1) = 0, G0(−1) = 0.
How much isG(0.3) ?
Exercise (8). Laplace transform Subexercise (A). Let
y0(t) + 5y(t) = 5t−1, y(0) = 6.
How much isY(1.1) ?
MCQ. A: 1.34591 B: 1.51199 C: 1.69857 D: 1.90817 E: 2.14364 Subexercise (B). Let
y0(t) + 5y(t) = 5t−1, y(0) = 6.
If
Y(s) =X
k,n
Ak,n
(s−αk)n, Ak,n6= 0, α1> α2 >· · · , then how much isA1,1 ?
MCQ. A:−75 B:−65 C: −45 D: −25 E: 15 Subexercise (C). Let
d dt~y =
−5 0 5 −6
~ y+
1 t
, ~y(0) = 6
6
.
How much isY1(1.1) ?
MCQ. A:{0.711137} B: {0.798892}C: {0.897475} D:{1.00822} E:{1.13264}
Exercise (9). Fourier transform, heat equation.
Subexercise (A). Letχ[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ[1.5,1.8](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.117896 B: 0.132444 C: 0.148787 D: 0.167148 E: 0.187774 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[1.5,1.8](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.00952042 B: 0.0106952 C: 0.012015 D: 0.0134977 E: 0.0151633 Exercise (10). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (3 + 4t)(4 + 3y) t0 = 3,∆t= 0.01.
d
dty(t) =f(t, y(t)), y(t0) =y0 = 4.
What is Heun’s prediction for y(t0+ ∆t)-re?
MCQ. A: 6.18181 B: 6.94464 C: 7.80161 D: 8.76433 E: 9.84585
Subexercise (E). (Taylor series) Let
f(t, y) = (3 + 4t)(4 + 3y) t0 = 3, d
dty(t) =f(t, y(t)), y(t0) =y0 = 4,
y(t0+ ∆t) =y0+c1∆t+c2∆t2+c3∆t3+O(∆t4).
How much isc2 ?
MCQ. A: 5428 B: 5429 C: 5430 D: 5432 E: 5434