StdNum: 11; DE, Teszt 1, Majus 9.
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty=−(y−5)(y−2)y2, y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 5 B: 6 C: 7 D: 8 E: 9
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: -2 B: -1 C: 0 D: 1 E: 2
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: 0 B: 5 C: ∞ D:−∞ E: 2
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: 2 D: 5 E: 0
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty=−(y−5)(y−2)y2, y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 5 B: 6 C: 7 D: 8 E: 9
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: -2 B: -1 C: 0 D: 1 E: 2
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: 0 B: 5 C: ∞ D:−∞ E: 2
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: 2 D: 5 E: 0
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
-0.2 0.0 0.2 0.4 0.6 -1
0 1 2 3 4 5 6
Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y =
f1 f2
=
(y1+ 2) (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: 0 B: 1 C: 2 D: 3 E: 4
Exercise (4). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−3 0 3 −7
, d
dt~y =A~y, ~y(0) = 1
3
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -8 B: -7 C: -6 D: -5 E: -4
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: 53 B: 73 C: 83 D: 103 E: 133
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: 74 B: 94 C: 114 D: 134 E: 154
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.0112051 B: 0.0125878 C: 0.0141411 D: 0.0158862 E: 0.0178465
Exercise (5). Hom.Lin. DE. Overdamped oscillator Let
¨
y+ay˙+by= 0, A=
0 1
−21 −10
, d
dt~y=A~y, ~y(0) = 1
3
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -8 B: -7 C: -6 D: -5 E: -4
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:−8 E: −6
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:−52 B:−32 C: −12 D: 12 E: 32
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: -0.0784357 B: -0.0881146 C: -0.098988 D: -0.111203 E: -0.124926
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(0.879A))13 ?
MCQ. A: 0.242555 B: 0.272486 C: 0.306111 D: 0.343885 E: 0.386321 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−2 0 2 −2
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(0.879A))21 ?
MCQ. A: 0.30306 B: 0.340458 C: 0.382471 D: 0.429667 E: 0.482688 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) + 4G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.301194 B: 0.338362 C: 0.380115 D: 0.427022 E: 0.479716 Subexercise (E).
G00(t) + 7G(t) =δ(t), G(−1) = 0, G0(−1) = 0.
How much isG(0.3) ?
Exercise (8). Laplace transform Subexercise (A). Let
y0(t) + 3y(t) = 5t−1, y(0) = 5.
How much isY(1.4) ?
MCQ. A: 1.38313 B: 1.5538 C: 1.74554 D: 1.96094 E: 2.20292 Subexercise (B). Let
y0(t) + 3y(t) = 5t−1, y(0) = 5.
If
Y(s) =X
k,n
Ak,n
(s−αk)n, Ak,n6= 0, α1> α2 >· · · , then how much isA1,1 ?
MCQ. A:−89 B:−59 C: −49 D: −29 E:−19 Subexercise (C). Let
d dt~y =
−2 0 2 −4
~ y+
1 t
, ~y(0) = 3
5
.
How much isY1(1.4) ?
MCQ. A:{0.770536} B: {0.865621}C: {0.972438} D:{1.09244} E:{1.22724}
Exercise (9). Fourier transform, heat equation.
Subexercise (A). Letχ[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ[0.7,2.2](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.397943 B: 0.447049 C: 0.502215 D: 0.564188 E: 0.633809 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[0.7,2.2](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.0226661 B: 0.0254631 C: 0.0286052 D: 0.0321351 E: 0.0361006 Exercise (10). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (2 + 2t)(2 + 2y) t0 = 2,∆t= 0.01.
d
dty(t) =f(t, y(t)), y(t0) =y0 = 2.
What is Heun’s prediction for y(t0+ ∆t)-re?
MCQ. A: 1.49573 B: 1.6803 C: 1.88765 D: 2.12059 E: 2.38227
Subexercise (E). (Taylor series) Let
f(t, y) = (2 + 2t)(2 + 2y) t0 = 2, d
dty(t) =f(t, y(t)), y(t0) =y0 = 2,
y(t0+ ∆t) =y0+c1∆t+c2∆t2+c3∆t3+O(∆t4).
How much isc2 ?
MCQ. A: 222 B: 224 C: 225 D: 226 E: 227