MA 303 Homework 1 (Homogeneous Linear Differential …MA 303 Homework 1 (Homogeneous Linear...

48
MA 303 Homework 1 (Homogeneous Linear Differential Equations) Hoon Hong 1st-order, 1 variable Problem: y' K 2 y =0 y(0) = K 3 1. Find the general solution: λ K 2=0 λ =2 y = C e 2 t 2. Find the particular solution: C = K 3 y = K 3 e 2 t 3. Sketch the particular solution: t y t 1 of 48

Transcript of MA 303 Homework 1 (Homogeneous Linear Differential …MA 303 Homework 1 (Homogeneous Linear...

MA 303Homework 1

(Homogeneous Linear Differential Equations)

Hoon Hong

1st-order, 1 variableProblem:

y'K2 y = 0

y(0) =K3

1. Find the general solution:λK2 = 0

λ = 2

y = C e2 t

2. Find the particular solution:C =K3

y =K3 e2 t

3. Sketch the particular solution:

t

y

t

1 of 48

Problem:y'K2 y = 0

y(0) = 4

1. Find the general solution:λK2 = 0

λ = 2

y = C e2 t

2. Find the particular solution:C = 4

y = 4 e2 t

3. Sketch the particular solution:

t

y

t

2 of 48

Problem:y'C3 y = 0

y(0) = 2

1. Find the general solution:λC3 = 0

λ =K3

y = C eK3 t

2. Find the particular solution:C = 2

y = 2 eK3 t

3. Sketch the particular solution:

t

y

t

3 of 48

Problem:y'C3 y = 0

y(0) =K2

1. Find the general solution:λC3 = 0

λ =K3

y = C eK3 t

2. Find the particular solution:C =K2

y =K2 eK3 t

3. Sketch the particular solution:

t

y

t

4 of 48

2nd-order, 1 variable: Real eigenvaluesProblem:

y''C3 y'C2 y = 0

y(0) =K3

y'(0) = 2

1. Find the general solution:

λ2C3 λC2 = 0

λ =K1, K2

y = C1 eKtCC2 e

K2 t

2. Find the particular solution:

y' =KC1 eKtK2 C2 e

K2 t

C1CC2 =K3

KC1K2 C2 = 2

C1 =K4, C2 = 1

y =K4 eKtCeK2 t

3. Sketch the particular solution:

t

y

t

5 of 48

Problem:y''K3 y'C2 y = 0

y(0) = 3

y'(0) = 1

1. Find the general solution:

λ2K3 λC2 = 0

λ = 2, 1

y = C1 e2 tCC2 e

t

2. Find the particular solution:y' = 2 C1 e

2 tCC2 et

C1CC2 = 3

2 C1CC2 = 1

C1 =K2, C2 = 5

y =K2 e2 tC5 et

3. Sketch the particular solution:

t

y

t

6 of 48

Problem:y''Cy'K6 y = 0

y(0) =K3

y'(0) = 4

1. Find the general solution:

λ2CλK6 = 0

λ = 2, K3

y = C1 e2 tCC2 e

K3 t

2. Find the particular solution:y' = 2 C1 e

2 tK3 C2 eK3 t

C1CC2 =K3

2 C1K3 C2 = 4

C1 =K1, C2 =K2

y =Ke2 tK2 eK3 t

3. Sketch the particular solution:

t

y

t

7 of 48

Problem:y''C5 y'C6 y = 0

y(0) = 3

y'(0) =K8

1. Find the general solution:

λ2C5 λC6 = 0

λ =K2, K3

y = C1 eK2 tCC2 e

K3 t

2. Find the particular solution:y' =K2 C1 e

K2 tK3 C2 eK3 t

C1CC2 = 3

K2 C1K3 C2 =K8

C1 = 1, C2 = 2

y = eK2 tC2 eK3 t

3. Sketch the particular solution:

t

y

t

8 of 48

Problem:y''C7 y'C12 y = 0

y(0) = 6

y'(0) =K22

1. Find the general solution:

λ2C7 λC12 = 0

λ =K3, K4

y = C1 eK3 tCC2 e

K4 t

2. Find the particular solution:y' =K3 C1 e

K3 tK4 C2 eK4 t

C1CC2 = 6

K3 C1K4 C2 =K22

C1 = 2, C2 = 4

y = 2 eK3 tC4 eK4 t

3. Sketch the particular solution:

t

y

t

9 of 48

Problem:y''K5 y'C6 y = 0

y(0) = 2

y'(0) = 1

1. Find the general solution:

λ2K5 λC6 = 0

λ = 3, 2

y = C1 e3 tCC2 e

2 t

2. Find the particular solution:y' = 3 C1 e

3 tC2 C2 e2 t

C1CC2 = 2

3 C1C2 C2 = 1

C1 =K3, C2 = 5

y =K3 e3 tC5 e2 t

3. Sketch the particular solution:

t

y

t

10 of 48

Problem:y''Cy'K2 y = 0

y(0) =K4

y'(0) = 5

1. Find the general solution:

λ2CλK2 = 0

λ = 1, K2

y = C1 etCC2 e

K2 t

2. Find the particular solution:y' = C1 e

tK2 C2 eK2 t

C1CC2 =K4

C1K2 C2 = 5

C1 =K1, C2 =K3

y =KetK3 eK2 t

3. Sketch the particular solution:

t

y

t

11 of 48

Problem:y''C5 y'C6 y = 0

y(0) =K1

y'(0) = 1

1. Find the general solution:

λ2C5 λC6 = 0

λ =K2, K3

y = C1 eK2 tCC2 e

K3 t

2. Find the particular solution:y' =K2 C1 e

K2 tK3 C2 eK3 t

C1CC2 =K1

K2 C1K3 C2 = 1

C1 =K2, C2 = 1

y =K2 eK2 tCeK3 t

3. Sketch the particular solution:

t

y

t

12 of 48

2nd-order, 1 variable: Non-real eigenvaluesProblem:

y''K4 y'C68 y = 0

y(0) = 1

y'(0) = 10

1. Find the general solution:

λ2K4 λC68 = 0

λ = 2C8 I, 2K8 I

y = e2 t C1 cos 8 t CC2 sin 8 t

2. Find the particular solution:

y' = 2 e2 t C1 cos 8 t CC2 sin 8 t Ce2 t K8 C1 sin 8 t C8 C2 cos 8 t

C1 = 1

2 C1C8 C2 = 10

C2 = 1

y = e2 t cos 8 t Csin 8 t

3. Sketch the particular solution:

t

y

t

13 of 48

Problem:y''C2 y'C50 y = 0

y(0) = 2

y'(0) = 19

1. Find the general solution:

λ2C2 λC50 = 0

λ =K1C7 I, K1K7 I

y = eKt C1 cos 7 t CC2 sin 7 t

2. Find the particular solution:y' =KeKt C1 cos 7 t CC2 sin 7 t CeKt K7 C1 sin 7 t C7 C2 cos 7 t

C1 = 2

KC1C7 C2 = 19

C2 = 3

y = eKt 2 cos 7 t C3 sin 7 t

3. Sketch the particular solution:

t

y

t

14 of 48

Problem:y''C64 y = 0

y(0) = 2

y'(0) = 8

1. Find the general solution:

λ2C64 = 0

λ = 8 I, K8 I

y = C1 cos 8 t CC2 sin 8 t

2. Find the particular solution:y' =K8 C1 sin 8 t C8 C2 cos 8 t

C1 = 2

8 C2 = 8

C2 = 1

y = 2 cos 8 t Csin 8 t

3. Sketch the particular solution:

t

y

t

15 of 48

Problem:y''C2 y'C82 y = 0

y(0) = 0

y'(0) = 9

1. Find the general solution:

λ2C2 λC82 = 0

λ =K1C9 I, K1K9 I

y = eKt C1 cos 9 t CC2 sin 9 t

2. Find the particular solution:y' =KeKt C1 cos 9 t CC2 sin 9 t CeKt K9 C1 sin 9 t C9 C2 cos 9 t

C1 = 0

KC1C9 C2 = 9

C2 = 1

y = eKt sin 9 t

3. Sketch the particular solution:

t

y

t

16 of 48

Problem:y''K4 y'C125 y = 0

y(0) = 1

y'(0) = 35

1. Find the general solution:

λ2K4 λC125 = 0

λ = 2C11 I, 2K11 I

y = e2 t C1 cos 11 t CC2 sin 11 t

2. Find the particular solution:y' = 2 e2 t C1 cos 11 t CC2 sin 11 t Ce2 t K11 C1 sin 11 t C11 C2 cos 11 t

C1 = 1

2 C1C11 C2 = 35

C2 = 3

y = e2 t cos 11 t C3 sin 11 t

3. Sketch the particular solution:

t

y

t

17 of 48

Problem:y''C9 y = 0

y(0) = 1

y'(0) =K3

1. Find the general solution:

λ2C9 = 0

λ = 3 I, K3 I

y = C1 cos 3 t CC2 sin 3 t

2. Find the particular solution:y' =K3 C1 sin 3 t C3 C2 cos 3 t

C1 = 1

3 C2 =K3

C2 =K1

y = cos 3 t Ksin 3 t

3. Sketch the particular solution:

t

y

t

18 of 48

1st-order, 2 variables: IntroductionProblem:

y1' =Ky1C2 y2

y2' = 2 y1C2 y2

y1(0) = 3

y2(0) = 1

1. Find the general solution:K1Kλ v1C2 v2 = 0

2 v1C 2Kλ v2 = 0

λ2KλK6 = 0

λ = 3

v1 = 2 C1

v2 = 4 C1

λ =K2

v1 = 2 C2

v2 =KC2

y1 = 2 C1 e3 tC2 C2 e

K2 t

y2 = 4 C1 e3 tKC2 e

K2 t

2. Find the particular solution:2 C1C2 C2 = 3

4 C1KC2 = 1

C1 =12

, C2 = 1

y1 = e3 tC2 eK2 t

y2 = 2 e3 tKeK2 t

19 of 48

Problem:y1' = y1K2 y2

y2' =K2 y1K2 y2

y1(0) = 3

y2(0) = 1

1. Find the general solution:1Kλ v1K2 v2 = 0

K2 v1C K2Kλ v2 = 0

λ2CλK6 = 0

λ =K3

v1 =K2 C1

v2 =K4 C1

λ = 2

v1 =K2 C2

v2 = C2

y1 =K2 C1 eK3 tK2 C2 e

2 t

y2 =K4 C1 eK3 tCC2 e

2 t

2. Find the particular solution:K2 C1K2 C2 = 3

K4 C1CC2 = 1

C1 =K12

, C2 =K1

y1 = eK3 tC2 e2 t

y2 = 2 eK3 tKe2 t

20 of 48

Problem:y1' =K5 y1K2 y2

y2' = 2 y1

y1(0) = 1

y2(0) = 1

1. Find the general solution:K5Kλ v1K2 v2 = 0

Kλ v2C2 v1 = 0

λ2C5 λC4 = 0

λ =K1

v1 =K2 C1

v2 = 4 C1

λ =K4

v1 =K2 C2

v2 = C2

y1 =K2 C1 eKtK2 C2 e

K4 t

y2 = 4 C1 eKtCC2 e

K4 t

2. Find the particular solution:K2 C1K2 C2 = 1

4 C1CC2 = 1

C1 =12

, C2 =K1

y1 =KeKtC2 eK4 t

y2 = 2 eKtKeK4 t

21 of 48

Problem:y1' = 3 y1Ky2

y2' =K2 y1C4 y2

y1(0) = 1

y2(0) = 4

1. Find the general solution:3Kλ v1Kv2 = 0

K2 v1C 4Kλ v2 = 0

λ2K7 λC10 = 0

λ = 2

v1 =KC1

v2 =KC1

λ = 5

v1 =KC2

v2 = 2 C2

y1 =KC1 e2 tKC2 e

5 t

y2 =KC1 e2 tC2 C2 e

5 t

2. Find the particular solution:KC1KC2 = 1

KC1C2 C2 = 4

C1 =K2, C2 = 1

y1 = 2 e2 tKe5 t

y2 = 2 e2 tC2 e5 t

22 of 48

1st-order, 2 variables: Real eigenvalues. Opposite SignsProblem:

y' =K1 2

2 2 y

y(0) =3

1

1. Find the general solution:

Det K1Kλ 2

2 2Kλ= 0

λ2KλK6 = 0

λ = 3, v =2

4

λ =K2, v =2

K1

y = C1 2

4 e3 tCC2

2

K1 eK2 t

2. Sketch the general solution:

y1

y2

Saddle

3. Find the particular solution:

23 of 48

C1 2

4CC2

2

K1=

3

1

C1 =12

, C2 = 1

y =1

2 e3 tC

2

K1 eK2 t

4. Sketch the particular solution:

t

y

t

24 of 48

Problem:

y' =1 K2

K2 K2 y

y(0) =3

1

1. Find the general solution:

Det 1Kλ K2

K2 K2Kλ= 0

λ2CλK6 = 0

λ =K3, v =K2

K4

λ = 2, v =K2

1

y = C1 K2

K4 eK3 tCC2

K2

1 e2 t

2. Sketch the general solution:

y1

y2

Saddle

3. Find the particular solution:

C1 K2

K4CC2

K2

1=

3

1

C1 =K12

, C2 =K1

25 of 48

y =1

2 eK3 tC

2

K1 e2 t

4. Sketch the particular solution:

t

y

t

26 of 48

Problem:

y' =K2 2

2 1 y

y(0) =2

K6

1. Find the general solution:

Det K2Kλ 2

2 1Kλ= 0

λ2CλK6 = 0

λ = 2, v =2

4

λ =K3, v =2

K1

y = C1 2

4 e2 tCC2

2

K1 eK3 t

2. Sketch the general solution:

y1

y2

Saddle

3. Find the particular solution:

C1 2

4CC2

2

K1=

2

K6

C1 =K1, C2 = 2

27 of 48

y =K2

K4 e2 tC

4

K2 eK3 t

4. Sketch the particular solution:

t

y

t

28 of 48

1st-order, 2 variables: Real eigenvalues. Same SignsProblem:

y' =3 K1

K2 4 y

y(0) =1

4

1. Find the general solution:

Det 3Kλ K1

K2 4Kλ= 0

λ2K7 λC10 = 0

λ = 2, v =K1

K1

λ = 5, v =K1

2

y = C1 K1

K1 e2 tCC2

K1

2 e5 t

2. Sketch the general solution:

y1

y2

Unstable

3. Find the particular solution:

29 of 48

C1 K1

K1CC2

K1

2=

1

4

C1 =K2, C2 = 1

y =2

2 e2 tC

K1

2 e5 t

4. Sketch the particular solution:

t

y

t

30 of 48

Problem:

y' =K3 2

1 K2 y

y(0) =8

2

1. Find the general solution:

Det K3Kλ 2

1 K2Kλ= 0

λ2C5 λC4 = 0

λ =K1, v =2

2

λ =K4, v =2

K1

y = C1 2

2 eKtCC2

2

K1 eK4 t

2. Sketch the general solution:

y1

y2

Stable

3. Find the particular solution:

C1 2

2CC2

2

K1=

8

2

C1 = 2, C2 = 2

31 of 48

y =4

4 eKtC

4

K2 eK4 t

4. Sketch the particular solution:

t

y

t

32 of 48

Problem:

y' =2 1

2 3 y

y(0) =0

3

1. Find the general solution:

Det 2Kλ 1

2 3Kλ= 0

λ2K5 λC4 = 0

λ = 1, v =1

K1

λ = 4, v =1

2

y = C1 1

K1 etCC2

1

2 e4 t

2. Sketch the general solution:

y1

y2

Unstable

3. Find the particular solution:

C1 1

K1CC2

1

2=

0

3

C1 =K1, C2 = 1

33 of 48

y =K1

1 etC

1

2 e4 t

4. Sketch the particular solution:

t

y

t

34 of 48

Problem:

y' =2 K1

K1 2 y

y(0) =K3

1

1. Find the general solution:

Det 2Kλ K1

K1 2Kλ= 0

λ2K4 λC3 = 0

λ = 1, v =K1

K1

λ = 3, v =K1

1

y = C1 K1

K1 etCC2

K1

1 e3 t

2. Sketch the general solution:

y1

y2

Unstable

3. Find the particular solution:

C1 K1

K1CC2

K1

1=

K3

1

C1 = 1, C2 = 2

35 of 48

y =K1

K1 etC

K2

2 e3 t

4. Sketch the particular solution:

t

y

t

36 of 48

1st-order 2 variables: Non-real eigenvaluesProblem:

y' =K2 K10

10 K2 y

y(0) =1

K1

1. Find the general solution:

Det K2Kλ K10

10 K2Kλ= 0

λ2C4 λC104 = 0

λ =K2C10 I, v =K10

10 I

λ =K2K10 I, v =K10

K10 I

α =K2, β = 10, p =K10

0, q =

0

10

y = eK2 t C1 K10

0CC2

0

10 cos 10 t C C2

K10

0KC1

0

10 sin 10 t

2. Sketch the general solution:

y1

y2

Stable

37 of 48

3. Find the particular solution:

C1 K10

0CC2

0

10=

1

K1

C1 =K110

, C2 =K110

y = eK2 t 1

K1 cos 10 t C

1

1 sin 10 t

4. Sketch the particular solution:

t

y

t

38 of 48

Problem:

y' =K7 16

K8 9 y

y(0) =4

K4

1. Find the general solution:

Det K7Kλ 16

K8 9Kλ= 0

λ2K2 λC65 = 0

λ = 1C8 I, v =16

8C8 I

λ = 1K8 I, v =16

8K8 I

α = 1, β = 8, p =16

8, q =

0

8

y = et C1 16

8CC2

0

8 cos 8 t C C2

16

8KC1

0

8 sin 8 t

2. Sketch the general solution:

y1

y2

Unstable

3. Find the particular solution:

39 of 48

C1 16

8CC2

0

8=

4

K4

C1 =14

, C2 =K34

y = et 4

K4 cos 8 t C

K12

K8 sin 8 t

4. Sketch the particular solution:

t

y

t

40 of 48

Problem:

y' =2 10

K4 K2 y

y(0) =1

1

1. Find the general solution:

Det 2Kλ 10

K4 K2Kλ= 0

λ2C36 = 0

λ = 6 I, v =10

K2C6 I

λ =K6 I, v =10

K2K6 I

α = 0, β = 6, p =10

K2, q =

0

6

y = C1 10

K2CC2

0

6 cos 6 t C C2

10

K2KC1

0

6 sin 6 t

2. Sketch the general solution:

y1

y2

center

3. Find the particular solution:

41 of 48

C1 10

K2CC2

0

6=

1

1

C1 =110

, C2 =15

y =1

1 cos 6 t C

2

K1 sin 6 t

4. Sketch the particular solution:

t

y

t

42 of 48

Problem:

y' =1 K4

4 1 y

y(0) =1

K1

1. Find the general solution:

Det 1Kλ K4

4 1Kλ= 0

λ2K2 λC17 = 0

λ = 1C4 I, v =K4

4 I

λ = 1K4 I, v =K4

K4 I

α = 1, β = 4, p =K4

0, q =

0

4

y = et C1 K4

0CC2

0

4 cos 4 t C C2

K4

0KC1

0

4 sin 4 t

2. Sketch the general solution:

y1

y2

Unstable

3. Find the particular solution:

43 of 48

C1 K4

0CC2

0

4=

1

K1

C1 =K14

, C2 =K14

y = et 1

K1 cos 4 t C

1

1 sin 4 t

4. Sketch the particular solution:

t

y

t

44 of 48

Problem:

y' =K14 24

K12 10 y

y(0) =4

K4

1. Find the general solution:

Det K14Kλ 24

K12 10Kλ= 0

λ2C4 λC148 = 0

λ =K2C12 I, v =24

12C12 I

λ =K2K12 I, v =24

12K12 I

α =K2, β = 12, p =24

12, q =

0

12

y = eK2 t C1 24

12CC2

0

12 cos 12 t C C2

24

12KC1

0

12 sin 12 t

2. Sketch the general solution:

y1

y2

Stable

3. Find the particular solution:

45 of 48

C1 24

12CC2

0

12=

4

K4

C1 =16

, C2 =K12

y = eK2 t 4

K4 cos 12 t C

K12

K8 sin 12 t

4. Sketch the particular solution:

t

y

t

46 of 48

Problem:

y' =6 K10

4 K6 y

y(0) =2

1

1. Find the general solution:

Det 6Kλ K10

4 K6Kλ= 0

λ2C4 = 0

λ = 2 I, v =K10

K6C2 I

λ =K2 I, v =K10

K6K2 I

α = 0, β = 2, p =K10

K6, q =

0

2

y = C1 K10

K6CC2

0

2 cos 2 t C C2

K10

K6KC1

0

2 sin 2 t

2. Sketch the general solution:

y1

y2

center

3. Find the particular solution:

47 of 48

C1 K10

K6CC2

0

2=

2

1

C1 =K15

, C2 =K110

y =2

1 cos 2 t C

1

1 sin 2 t

4. Sketch the particular solution:

t

y

t

48 of 48