StdNum: 17; DE, Teszt 1, Majus 9.
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−4)(y−3)(y−1), y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 8 B: 9 C: 10 D: 11 E: 12
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: 3 B: 4 C: 5 D: 6 E: 7
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:−∞ B: 3 C:∞ D: 1 E: 4
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: 3 B: −∞C: 4 D: 1 E: ∞
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−4)(y−3)(y−1), y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 8 B: 9 C: 10 D: 11 E: 12
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: 3 B: 4 C: 5 D: 6 E: 7
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:−∞ B: 3 C:∞ D: 1 E: 4
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: 3 B: −∞C: 4 D: 1 E: ∞
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y=
f1 f2
=
(y1+ 4)y2 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: 3 B: 4 C: 5 D: 6 E: 7
Exercise (4). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−2 0 2 −5
, d
dt~y =A~y, ~y(0) = 2
2
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -7 B: -6 C: -5 D: -4 E: -3
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:−12 B: 12 C: 32 D: 52 E: 72
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: 23 C: 43 D: 53 E: 73
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.0281656 B: 0.0316412 C: 0.0355458 D: 0.0399321 E: 0.0448597
Exercise (5). Hom.Lin. DE. Overdamped oscillator Let
¨
y+ay˙+by= 0, A=
0 1
−10 −7
, d
dt~y=A~y, ~y(0) = 2
2
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -7 B: -6 C: -5 D: -4 E: -3
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:−10 B: −9 C: −8 D:−7 E: −5
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:−4 B:−2 C: 0 D: 1 E: 2
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: -0.140828 B: -0.158206 C: -0.177729 D: -0.199661 E: -0.224299
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(1.452A))13 ?
MCQ. A: 0.743533 B: 0.835284 C: 0.938359 D: 1.05415 E: 1.18423 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(1.452A))21 ?
MCQ. A: 0.426838 B: 0.47951 C: 0.538682 D: 0.605155 E: 0.679831 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) + 5G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.176803 B: 0.19862 C: 0.22313 D: 0.250664 E: 0.281596 Subexercise (E).
G00(t) + 6G(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(1.2) ?
MCQ. A: 1.31708 B: 1.4796 C: 1.66219 D: 1.8673 E: 2.09773 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:−4 B:−3 C: −2 D: 0 E: 2 Subexercise (C). Let
d dt~y =
−7 0 7 −3
~ y+
1 t
, ~y(0) = 4
4
.
How much isY1(1.2) ?
MCQ. A:{0.415747} B: {0.467051}C: {0.524685} D:{0.589431} E:{0.662167}
Exercise (9). Fourier transform, heat equation.
Subexercise (A). Letχ[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ[0.5,1.9](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.277295 B: 0.311513 C: 0.349953 D: 0.393138 E: 0.441651 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[0.5,1.9](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.0282598 B: 0.031747 C: 0.0356646 D: 0.0400657 E: 0.0450098 Exercise (10). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (3 + 3t)(3 + 3y) t0 = 3,∆t= 0.01.
d
dty(t) =f(t, y(t)), y(t0) =y0 = 3.
What is Heun’s prediction for y(t0+ ∆t)-re?
MCQ. A: 3.72547 B: 4.18519 C: 4.70165 D: 5.28183 E: 5.93361
Subexercise (E). (Taylor series) Let
f(t, y) = (3 + 3t)(3 + 3y) t0 = 3, d
dty(t) =f(t, y(t)), y(t0) =y0 = 3,
y(t0+ ∆t) =y0+c1∆t+c2∆t2+c3∆t3+O(∆t4).
How much isc2 ?
MCQ. A: 2606 B: 2607 C: 2608 D: 2610 E: 2612