C1 - Homogeneous first-order linear ODE


Example 1

C1 - Homogeneous first-order linear ODE (ver. 1)

Find the general solution to the given ODE.

\[-30 \, {y} = 5 \, {y'}\]

Answer.

\[y= k e^{\left(-6 \, t\right)}\]


Example 2

C1 - Homogeneous first-order linear ODE (ver. 2)

Find the general solution to the given ODE.

\[-15 \, {y} - 3 \, {y'} = 0\]

Answer.

\[y= k e^{\left(-5 \, t\right)}\]


Example 3

C1 - Homogeneous first-order linear ODE (ver. 3)

Find the general solution to the given ODE.

\[-5 \, {y'} + 40 \, {y} = 0\]

Answer.

\[y= k e^{\left(8 \, t\right)}\]


Example 4

C1 - Homogeneous first-order linear ODE (ver. 4)

Find the general solution to the given ODE.

\[5 \, {y'} = -15 \, {y}\]

Answer.

\[y= k e^{\left(-3 \, t\right)}\]


Example 5

C1 - Homogeneous first-order linear ODE (ver. 5)

Find the general solution to the given ODE.

\[0 = 4 \, {y'} + 20 \, {y}\]

Answer.

\[y= k e^{\left(-5 \, t\right)}\]


Example 6

C1 - Homogeneous first-order linear ODE (ver. 6)

Find the general solution to the given ODE.

\[-4 \, {y'} = 36 \, {y}\]

Answer.

\[y= k e^{\left(-9 \, t\right)}\]


Example 7

C1 - Homogeneous first-order linear ODE (ver. 7)

Find the general solution to the given ODE.

\[-35 \, {y} + 5 \, {y'} = 0\]

Answer.

\[y= k e^{\left(7 \, t\right)}\]


Example 8

C1 - Homogeneous first-order linear ODE (ver. 8)

Find the general solution to the given ODE.

\[45 \, {y} - 5 \, {y'} = 0\]

Answer.

\[y= k e^{\left(9 \, t\right)}\]


Example 9

C1 - Homogeneous first-order linear ODE (ver. 9)

Find the general solution to the given ODE.

\[-2 \, {y'} = -6 \, {y}\]

Answer.

\[y= k e^{\left(3 \, t\right)}\]


Example 10

C1 - Homogeneous first-order linear ODE (ver. 10)

Find the general solution to the given ODE.

\[0 = 20 \, {y} + 5 \, {y'}\]

Answer.

\[y= k e^{\left(-4 \, t\right)}\]


Example 11

C1 - Homogeneous first-order linear ODE (ver. 11)

Find the general solution to the given ODE.

\[0 = 3 \, {y'} + 15 \, {y}\]

Answer.

\[y= k e^{\left(-5 \, t\right)}\]


Example 12

C1 - Homogeneous first-order linear ODE (ver. 12)

Find the general solution to the given ODE.

\[18 \, {y} = -3 \, {y'}\]

Answer.

\[y= k e^{\left(-6 \, t\right)}\]


Example 13

C1 - Homogeneous first-order linear ODE (ver. 13)

Find the general solution to the given ODE.

\[0 = -2 \, {y'} - 16 \, {y}\]

Answer.

\[y= k e^{\left(-8 \, t\right)}\]


Example 14

C1 - Homogeneous first-order linear ODE (ver. 14)

Find the general solution to the given ODE.

\[6 \, {y} = 3 \, {y'}\]

Answer.

\[y= k e^{\left(2 \, t\right)}\]


Example 15

C1 - Homogeneous first-order linear ODE (ver. 15)

Find the general solution to the given ODE.

\[4 \, {y'} = -28 \, {y}\]

Answer.

\[y= k e^{\left(-7 \, t\right)}\]


Example 16

C1 - Homogeneous first-order linear ODE (ver. 16)

Find the general solution to the given ODE.

\[12 \, {y} = -2 \, {y'}\]

Answer.

\[y= k e^{\left(-6 \, t\right)}\]


Example 17

C1 - Homogeneous first-order linear ODE (ver. 17)

Find the general solution to the given ODE.

\[15 \, {y} = -3 \, {y'}\]

Answer.

\[y= k e^{\left(-5 \, t\right)}\]


Example 18

C1 - Homogeneous first-order linear ODE (ver. 18)

Find the general solution to the given ODE.

\[-5 \, {y'} = -20 \, {y}\]

Answer.

\[y= k e^{\left(4 \, t\right)}\]


Example 19

C1 - Homogeneous first-order linear ODE (ver. 19)

Find the general solution to the given ODE.

\[3 \, {y'} = 27 \, {y}\]

Answer.

\[y= k e^{\left(9 \, t\right)}\]


Example 20

C1 - Homogeneous first-order linear ODE (ver. 20)

Find the general solution to the given ODE.

\[0 = -5 \, {y'} + 30 \, {y}\]

Answer.

\[y= k e^{\left(6 \, t\right)}\]


Example 21

C1 - Homogeneous first-order linear ODE (ver. 21)

Find the general solution to the given ODE.

\[5 \, {y'} = -5 \, {y}\]

Answer.

\[y= k e^{\left(-t\right)}\]


Example 22

C1 - Homogeneous first-order linear ODE (ver. 22)

Find the general solution to the given ODE.

\[-5 \, {y'} - 25 \, {y} = 0\]

Answer.

\[y= k e^{\left(-5 \, t\right)}\]


Example 23

C1 - Homogeneous first-order linear ODE (ver. 23)

Find the general solution to the given ODE.

\[-5 \, {y'} = 10 \, {y}\]

Answer.

\[y= k e^{\left(-2 \, t\right)}\]


Example 24

C1 - Homogeneous first-order linear ODE (ver. 24)

Find the general solution to the given ODE.

\[0 = -5 \, {y'} - 40 \, {y}\]

Answer.

\[y= k e^{\left(-8 \, t\right)}\]


Example 25

C1 - Homogeneous first-order linear ODE (ver. 25)

Find the general solution to the given ODE.

\[-21 \, {y} = -3 \, {y'}\]

Answer.

\[y= k e^{\left(7 \, t\right)}\]


Example 26

C1 - Homogeneous first-order linear ODE (ver. 26)

Find the general solution to the given ODE.

\[28 \, {y} + 4 \, {y'} = 0\]

Answer.

\[y= k e^{\left(-7 \, t\right)}\]


Example 27

C1 - Homogeneous first-order linear ODE (ver. 27)

Find the general solution to the given ODE.

\[16 \, {y} = 2 \, {y'}\]

Answer.

\[y= k e^{\left(8 \, t\right)}\]


Example 28

C1 - Homogeneous first-order linear ODE (ver. 28)

Find the general solution to the given ODE.

\[-2 \, {y'} + 18 \, {y} = 0\]

Answer.

\[y= k e^{\left(9 \, t\right)}\]


Example 29

C1 - Homogeneous first-order linear ODE (ver. 29)

Find the general solution to the given ODE.

\[-20 \, {y} = 4 \, {y'}\]

Answer.

\[y= k e^{\left(-5 \, t\right)}\]


Example 30

C1 - Homogeneous first-order linear ODE (ver. 30)

Find the general solution to the given ODE.

\[4 \, {y'} = -28 \, {y}\]

Answer.

\[y= k e^{\left(-7 \, t\right)}\]


Example 31

C1 - Homogeneous first-order linear ODE (ver. 31)

Find the general solution to the given ODE.

\[0 = 5 \, {y'} - 30 \, {y}\]

Answer.

\[y= k e^{\left(6 \, t\right)}\]


Example 32

C1 - Homogeneous first-order linear ODE (ver. 32)

Find the general solution to the given ODE.

\[-18 \, {y} + 3 \, {y'} = 0\]

Answer.

\[y= k e^{\left(6 \, t\right)}\]


Example 33

C1 - Homogeneous first-order linear ODE (ver. 33)

Find the general solution to the given ODE.

\[2 \, {y'} - 8 \, {y} = 0\]

Answer.

\[y= k e^{\left(4 \, t\right)}\]


Example 34

C1 - Homogeneous first-order linear ODE (ver. 34)

Find the general solution to the given ODE.

\[-40 \, {y} = 5 \, {y'}\]

Answer.

\[y= k e^{\left(-8 \, t\right)}\]


Example 35

C1 - Homogeneous first-order linear ODE (ver. 35)

Find the general solution to the given ODE.

\[4 \, {y'} - 32 \, {y} = 0\]

Answer.

\[y= k e^{\left(8 \, t\right)}\]


Example 36

C1 - Homogeneous first-order linear ODE (ver. 36)

Find the general solution to the given ODE.

\[0 = -12 \, {y} + 3 \, {y'}\]

Answer.

\[y= k e^{\left(4 \, t\right)}\]


Example 37

C1 - Homogeneous first-order linear ODE (ver. 37)

Find the general solution to the given ODE.

\[21 \, {y} = 3 \, {y'}\]

Answer.

\[y= k e^{\left(7 \, t\right)}\]


Example 38

C1 - Homogeneous first-order linear ODE (ver. 38)

Find the general solution to the given ODE.

\[4 \, {y} = -2 \, {y'}\]

Answer.

\[y= k e^{\left(-2 \, t\right)}\]


Example 39

C1 - Homogeneous first-order linear ODE (ver. 39)

Find the general solution to the given ODE.

\[-2 \, {y'} = -4 \, {y}\]

Answer.

\[y= k e^{\left(2 \, t\right)}\]


Example 40

C1 - Homogeneous first-order linear ODE (ver. 40)

Find the general solution to the given ODE.

\[-16 \, {y} = 4 \, {y'}\]

Answer.

\[y= k e^{\left(-4 \, t\right)}\]


Example 41

C1 - Homogeneous first-order linear ODE (ver. 41)

Find the general solution to the given ODE.

\[-3 \, {y'} + 27 \, {y} = 0\]

Answer.

\[y= k e^{\left(9 \, t\right)}\]


Example 42

C1 - Homogeneous first-order linear ODE (ver. 42)

Find the general solution to the given ODE.

\[0 = -3 \, {y'} + 27 \, {y}\]

Answer.

\[y= k e^{\left(9 \, t\right)}\]


Example 43

C1 - Homogeneous first-order linear ODE (ver. 43)

Find the general solution to the given ODE.

\[4 \, {y'} + 28 \, {y} = 0\]

Answer.

\[y= k e^{\left(-7 \, t\right)}\]


Example 44

C1 - Homogeneous first-order linear ODE (ver. 44)

Find the general solution to the given ODE.

\[-4 \, {y'} = 4 \, {y}\]

Answer.

\[y= k e^{\left(-t\right)}\]


Example 45

C1 - Homogeneous first-order linear ODE (ver. 45)

Find the general solution to the given ODE.

\[0 = -12 \, {y} - 2 \, {y'}\]

Answer.

\[y= k e^{\left(-6 \, t\right)}\]


Example 46

C1 - Homogeneous first-order linear ODE (ver. 46)

Find the general solution to the given ODE.

\[0 = 5 \, {y'} + 40 \, {y}\]

Answer.

\[y= k e^{\left(-8 \, t\right)}\]


Example 47

C1 - Homogeneous first-order linear ODE (ver. 47)

Find the general solution to the given ODE.

\[0 = 8 \, {y} + 2 \, {y'}\]

Answer.

\[y= k e^{\left(-4 \, t\right)}\]


Example 48

C1 - Homogeneous first-order linear ODE (ver. 48)

Find the general solution to the given ODE.

\[-20 \, {y} = -4 \, {y'}\]

Answer.

\[y= k e^{\left(5 \, t\right)}\]


Example 49

C1 - Homogeneous first-order linear ODE (ver. 49)

Find the general solution to the given ODE.

\[-2 \, {y'} - 6 \, {y} = 0\]

Answer.

\[y= k e^{\left(-3 \, t\right)}\]


Example 50

C1 - Homogeneous first-order linear ODE (ver. 50)

Find the general solution to the given ODE.

\[0 = 3 \, {y} - 3 \, {y'}\]

Answer.

\[y= k e^{t}\]