T 4. Tests for general and self-studying

ì

y

ì

ïìdx

t

ïdx

x

ï

,

ïdx

ï

=

,

=

ï

=

,

2

ï

ï

-

ï

+

x(0) = 2,

ï

y

ïdt

(x

y)

ï

x

y

ïdt

2.

ï

3.

ïdt

.

1. í

í

í

ï

t

ï

x

ï

y

y(0) = 4.

ïdy

.

ïdy

=

.

ïdy

=

;

ï

=-

ï

ï

2

ï

x

ï

(x

- y)

ï

x + y

îïdt

îïdt

îïdt

Solve the simultaneous differential equations by the method of eliminating

or generalized Euler’s method

dx

3x y,

dx

y

,

dx

x 4y,

4.

5.

2

6.

dt

dt

dt

dy

20x 6 y.

dy

2x.

dy

y 3x.

dt

dt

dt

dx

2x y,

dx

4x 6 y,

dx

y 2e

t

,

7.

dt

8.

dt

9.

dt

dy

3x 4y.

dy

2x 3y t.

dy

x t

2

.

dt

dt

dt

dx

2x y,

dx

2x y cos t,

10. dt

11.

dt

dy

2y x 5e

t

sin t.

dy

x 2 sin t.

dt

dt

eC1t2 ,

1

e-C1t2

Answers

1. x = C

y =

.

2. x2 - y2 = C ,

2x +(x - y)2 = C

2

.

3.

2

2C1C2

1

x =

t

+ 2, y =

2t

+ 4 .

4.

x C e2t C

et , y 5C e2t

4C

et . 5.

x C cost C

2

sin t,

3

3

1

2

1

2

1

y 2C sin t 2C

2

cost.

6.

x et (C cos 2

3t C

2

sin 2

3t),

y

3

et (C

2

cos2

3t

1

1

2

C sin 2

3t). 7.

x C et C

e5t ,

y C et

3C

e5t

.

8.

x C

C

e7t

3

(7t 2)t,

1

1

2

1

2

1

2

49

y

2

C1

1

C2e7t

1

(14t2

3t 1).

9.

x = C1et

+C2e-t +tet -t2 -2,

3

2

49

4.2.15.

a11 3,

a12 13,

a21 1,

a22 1.

4.2.16.

a11 6,

a12 5,

a21 1,

a22 2 .

4.2.17.

a11 7,

a12 1,

a21 1,

a22 5 .

4.2.18.

a11 3,

a12 2,

a21 4,

a22 1 .

4.2.19.

a11 2,

a12 2,

a21 2,

a22 5 .

4.2.20.

a11 8,

a12 4,

a21 1,

a22 4 .

4.2.21.

a11 5,

a12 5,

a21 5,

a22 1 .

4.2.22.

a11 7,

a12 2,

a21 3,

a22 2 .

4.2.23.

a11 9,

a12 9,

a21 1,

a22 3.

4.2.24.

a11 4,

a12 2,

a21 5,

a22 2 .

4.2.25.

a11 6,

a12 2,

a21 5,

a22 3.

4.2.26.

a11 4,

a12 3,

a21 3,

a22 2 .

4.2.27.

a11 5,

a12 1,

a21 2,

a22 3 .

4.2.28.

a11 7,

a12 6,

a21 4,

a22 3 .

4.2.29.

a11 3,

a12 2,

a21 2,

a22 1 .

4.2.30.

a11 7,

a12 3,

a21 6,

a22 1.