StdNum: 21; DE, Teszt 1, Majus 9.
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−2)y(y+ 2), y(0) =−2.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: -3 B: -2 C: -1 D: 0 E: 1
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: 4 B: 5 C: 6 D: 7 E: 8
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: 2 B: 0 C: −2 D:∞ 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: 2 B: 0 C: −∞ D:∞ E:−2
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−2)y(y+ 2), y(0) =−2.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: -3 B: -2 C: -1 D: 0 E: 1
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: 4 B: 5 C: 6 D: 7 E: 8
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: 2 B: 0 C: −2 D:∞ 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: 2 B: 0 C: −∞ D:∞ E:−2
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y =
f1 f2
=
(y1+ 5) (y2+ 2)
−y1
.
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=
−1 0 1 −3
, d
dt~y =A~y, ~y(0) = 3
4
.
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: 2 B: 4 C: 6 D: 7 E: 8
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: 12 B: 32 C: 52 D: 72 E: 92
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.0917577 B: 0.103081 C: 0.115801 D: 0.13009 E: 0.146144
Exercise (5). Hom.Lin. DE. Overdamped oscillator Let
¨
y+ay˙+by= 0, A=
0 1
−3 −4
, d
dt~y=A~y, ~y(0) = 3
4
.
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:−4 B:−2 C: 0 D: 1 E: 2
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:−112 B: −92 C:−72 D:−52 E:−32
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: -0.275273 B: -0.309242 C: -0.347402 D: -0.390271 E: -0.438431
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(1.055A))13 ?
MCQ. A: 0.495382 B: 0.556513 C: 0.625186 D: 0.702334 E: 0.789002 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−1 0 3 −1
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(1.055A))21 ?
MCQ. A: 0.777302 B: 0.873221 C: 0.980977 D: 1.10203 E: 1.23802 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) + 6G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.116591 B: 0.130979 C: 0.147142 D: 0.165299 E: 0.185697 Subexercise (E).
G00(t) + 4G(t) =δ(t), G(−1) = 0, G0(−1) = 0.
How much isG(0.3) ?
Exercise (8). Laplace transform Subexercise (A). Let
y0(t) + 4y(t) = 3t+ 1, y(0) = 5.
How much isY(1.) ?
MCQ. A: 1.42628 B: 1.60228 C: 1.8 D: 2.02212 E: 2.27165 Subexercise (B). Let
y0(t) + 4y(t) = 3t+ 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:−163 B: −161 C: 161 D: 163 E: 165 Subexercise (C). Let
d dt~y =
−6 0 6 −3
~ y+
1 t
, ~y(0) = 4
7
.
How much isY1(1.) ?
MCQ. A:{0.448471} B: {0.503812}C: {0.565983} D:{0.635825} E:{0.714286}
Exercise (9). Fourier transform, heat equation.
Subexercise (A). Letχ[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ[1.,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: 0.262265 B: 0.294629 C: 0.330986 D: 0.37183 E: 0.417714 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[1.,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: 0.0211787 B: 0.0237922 C: 0.0267281 D: 0.0300264 E: 0.0337316 Exercise (10). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (2 + 4t)(4 + 2y) t0 = 2,∆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: 3.75442 B: 4.21772 C: 4.73819 D: 5.32288 E: 5.97972
Subexercise (E). (Taylor series) Let
f(t, y) = (2 + 4t)(4 + 2y) t0 = 2, 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: 1221 B: 1222 C: 1224 D: 1226 E: 1227