F2: Separation of variables


Example 1

F2: Separation of variables (ver. 1)

Find the solution to the given IVP.

\[-{\left(4 \, t + 3\right)} {y} = {y'} \hspace{1em} y( 0 ) = -2 \, e^{5}\]

Answer.

\[{y} = -2 \, e^{\left(-2 \, t^{2} - 3 \, t + 5\right)}\]


Example 2

F2: Separation of variables (ver. 2)

Find the solution to the given IVP.

\[0 = 3 \, {y} \cos\left(t\right) + {y'} \hspace{1em} y( 0 ) = 2\]

Answer.

\[{y} = 2 \, e^{\left(-3 \, \sin\left(t\right)\right)}\]


Example 3

F2: Separation of variables (ver. 3)

Find the solution to the given IVP.

\[t {y'} = 3 \, {y} \hspace{1em} y( 2 ) = 32\]

Answer.

\[{y} = 4 \, t^{3}\]


Example 4

F2: Separation of variables (ver. 4)

Find the solution to the given IVP.

\[-{y} \sin\left(t\right) - {y'} = 0 \hspace{1em} y( 0 ) = 3 \, e\]

Answer.

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


Example 5

F2: Separation of variables (ver. 5)

Find the solution to the given IVP.

\[t {y'} + 3 \, {y} = 0 \hspace{1em} y( 2 ) = -\frac{1}{4}\]

Answer.

\[{y} = -\frac{2}{t^{3}}\]


Example 6

F2: Separation of variables (ver. 6)

Find the solution to the given IVP.

\[-2 \, {y} \sin\left(t\right) - {y'} = 0 \hspace{1em} y( 0 ) = -e^{2}\]

Answer.

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


Example 7

F2: Separation of variables (ver. 7)

Find the solution to the given IVP.

\[0 = -3 \, {\left(2 \, t + 1\right)} {y} + {y'} \hspace{1em} y( 0 ) = -2 \, e^{3}\]

Answer.

\[{y} = -2 \, e^{\left(3 \, t^{2} + 3 \, t + 3\right)}\]


Example 8

F2: Separation of variables (ver. 8)

Find the solution to the given IVP.

\[{y'} = -2 \, {y} \cos\left(t\right) \hspace{1em} y( 0 ) = -3\]

Answer.

\[{y} = -3 \, e^{\left(-2 \, \sin\left(t\right)\right)}\]


Example 9

F2: Separation of variables (ver. 9)

Find the solution to the given IVP.

\[-2 \, {y} \sin\left(t\right) - {y'} = 0 \hspace{1em} y( 0 ) = 3 \, e^{2}\]

Answer.

\[{y} = 3 \, e^{\left(2 \, \cos\left(t\right)\right)}\]


Example 10

F2: Separation of variables (ver. 10)

Find the solution to the given IVP.

\[{y} \sin\left(t\right) = {y'} \hspace{1em} y( 0 ) = e^{\left(-1\right)}\]

Answer.

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


Example 11

F2: Separation of variables (ver. 11)

Find the solution to the given IVP.

\[t {y'} + 2 \, {y} = 0 \hspace{1em} y( -1 ) = -1\]

Answer.

\[{y} = -\frac{1}{t^{2}}\]


Example 12

F2: Separation of variables (ver. 12)

Find the solution to the given IVP.

\[-3 \, {y} = t {y'} \hspace{1em} y( -4 ) = \frac{1}{16}\]

Answer.

\[{y} = -\frac{4}{t^{3}}\]


Example 13

F2: Separation of variables (ver. 13)

Find the solution to the given IVP.

\[t {y'} + 3 \, {y} = 0 \hspace{1em} y( -3 ) = \frac{2}{27}\]

Answer.

\[{y} = -\frac{2}{t^{3}}\]


Example 14

F2: Separation of variables (ver. 14)

Find the solution to the given IVP.

\[2 \, {y} = t {y'} \hspace{1em} y( -1 ) = -3\]

Answer.

\[{y} = -3 \, t^{2}\]


Example 15

F2: Separation of variables (ver. 15)

Find the solution to the given IVP.

\[-{\left(6 \, t - 1\right)} {y} = {y'} \hspace{1em} y( 0 ) = e^{2}\]

Answer.

\[{y} = e^{\left(-3 \, t^{2} + t + 2\right)}\]


Example 16

F2: Separation of variables (ver. 16)

Find the solution to the given IVP.

\[t {y'} = -3 \, {y} \hspace{1em} y( -4 ) = \frac{1}{16}\]

Answer.

\[{y} = -\frac{4}{t^{3}}\]


Example 17

F2: Separation of variables (ver. 17)

Find the solution to the given IVP.

\[-2 \, {y} = t {y'} \hspace{1em} y( -4 ) = -\frac{3}{16}\]

Answer.

\[{y} = -\frac{3}{t^{2}}\]


Example 18

F2: Separation of variables (ver. 18)

Find the solution to the given IVP.

\[-{y} \cos\left(t\right) = {y'} \hspace{1em} y( 0 ) = 2\]

Answer.

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


Example 19

F2: Separation of variables (ver. 19)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = 0 \hspace{1em} y( -3 ) = -54\]

Answer.

\[{y} = 2 \, t^{3}\]


Example 20

F2: Separation of variables (ver. 20)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} \hspace{1em} y( 4 ) = -64\]

Answer.

\[{y} = -t^{3}\]


Example 21

F2: Separation of variables (ver. 21)

Find the solution to the given IVP.

\[t {y'} + 2 \, {y} = 0 \hspace{1em} y( 3 ) = \frac{2}{9}\]

Answer.

\[{y} = \frac{2}{t^{2}}\]


Example 22

F2: Separation of variables (ver. 22)

Find the solution to the given IVP.

\[0 = -t {y'} - 2 \, {y} \hspace{1em} y( 1 ) = 4\]

Answer.

\[{y} = \frac{4}{t^{2}}\]


Example 23

F2: Separation of variables (ver. 23)

Find the solution to the given IVP.

\[t {y'} = -2 \, {y} \hspace{1em} y( 2 ) = \frac{1}{4}\]

Answer.

\[{y} = \frac{1}{t^{2}}\]


Example 24

F2: Separation of variables (ver. 24)

Find the solution to the given IVP.

\[t {y'} + 3 \, {y} = 0 \hspace{1em} y( 3 ) = -\frac{2}{27}\]

Answer.

\[{y} = -\frac{2}{t^{3}}\]


Example 25

F2: Separation of variables (ver. 25)

Find the solution to the given IVP.

\[t {y'} = 3 \, {y} \hspace{1em} y( -3 ) = 108\]

Answer.

\[{y} = -4 \, t^{3}\]


Example 26

F2: Separation of variables (ver. 26)

Find the solution to the given IVP.

\[0 = 3 \, {y} \sin\left(t\right) + {y'} \hspace{1em} y( 0 ) = -e^{3}\]

Answer.

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


Example 27

F2: Separation of variables (ver. 27)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = 0 \hspace{1em} y( -2 ) = 24\]

Answer.

\[{y} = -3 \, t^{3}\]


Example 28

F2: Separation of variables (ver. 28)

Find the solution to the given IVP.

\[0 = -t {y'} + 3 \, {y} \hspace{1em} y( -2 ) = -32\]

Answer.

\[{y} = 4 \, t^{3}\]


Example 29

F2: Separation of variables (ver. 29)

Find the solution to the given IVP.

\[{y'} = -2 \, {y} \cos\left(t\right) \hspace{1em} y( 0 ) = 3\]

Answer.

\[{y} = 3 \, e^{\left(-2 \, \sin\left(t\right)\right)}\]


Example 30

F2: Separation of variables (ver. 30)

Find the solution to the given IVP.

\[t {y'} = -3 \, {y} \hspace{1em} y( 3 ) = \frac{1}{27}\]

Answer.

\[{y} = \frac{1}{t^{3}}\]


Example 31

F2: Separation of variables (ver. 31)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = 0 \hspace{1em} y( 1 ) = -2\]

Answer.

\[{y} = -2 \, t^{3}\]


Example 32

F2: Separation of variables (ver. 32)

Find the solution to the given IVP.

\[-3 \, {y} \sin\left(t\right) = {y'} \hspace{1em} y( 0 ) = -3 \, e^{3}\]

Answer.

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


Example 33

F2: Separation of variables (ver. 33)

Find the solution to the given IVP.

\[2 \, {y} = t {y'} \hspace{1em} y( -4 ) = 48\]

Answer.

\[{y} = 3 \, t^{2}\]


Example 34

F2: Separation of variables (ver. 34)

Find the solution to the given IVP.

\[{y'} = 3 \, {y} \sin\left(t\right) \hspace{1em} y( 0 ) = -e^{\left(-3\right)}\]

Answer.

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


Example 35

F2: Separation of variables (ver. 35)

Find the solution to the given IVP.

\[0 = -t {y'} + 3 \, {y} \hspace{1em} y( 4 ) = 256\]

Answer.

\[{y} = 4 \, t^{3}\]


Example 36

F2: Separation of variables (ver. 36)

Find the solution to the given IVP.

\[-3 \, {y} \cos\left(t\right) - {y'} = 0 \hspace{1em} y( 0 ) = 3\]

Answer.

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


Example 37

F2: Separation of variables (ver. 37)

Find the solution to the given IVP.

\[2 \, {y} \sin\left(t\right) - {y'} = 0 \hspace{1em} y( 0 ) = -3 \, e^{\left(-2\right)}\]

Answer.

\[{y} = -3 \, e^{\left(-2 \, \cos\left(t\right)\right)}\]


Example 38

F2: Separation of variables (ver. 38)

Find the solution to the given IVP.

\[t {y'} = -2 \, {y} \hspace{1em} y( -3 ) = -\frac{4}{9}\]

Answer.

\[{y} = -\frac{4}{t^{2}}\]


Example 39

F2: Separation of variables (ver. 39)

Find the solution to the given IVP.

\[{y'} = -{y} \sin\left(t\right) \hspace{1em} y( 0 ) = 4 \, e\]

Answer.

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


Example 40

F2: Separation of variables (ver. 40)

Find the solution to the given IVP.

\[t {y'} + 2 \, {y} = 0 \hspace{1em} y( 1 ) = -3\]

Answer.

\[{y} = -\frac{3}{t^{2}}\]


Example 41

F2: Separation of variables (ver. 41)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} \hspace{1em} y( -1 ) = -4\]

Answer.

\[{y} = 4 \, t^{3}\]


Example 42

F2: Separation of variables (ver. 42)

Find the solution to the given IVP.

\[t {y'} = 3 \, {y} \hspace{1em} y( -3 ) = 54\]

Answer.

\[{y} = -2 \, t^{3}\]


Example 43

F2: Separation of variables (ver. 43)

Find the solution to the given IVP.

\[-3 \, {y} = t {y'} \hspace{1em} y( 3 ) = \frac{4}{27}\]

Answer.

\[{y} = \frac{4}{t^{3}}\]


Example 44

F2: Separation of variables (ver. 44)

Find the solution to the given IVP.

\[t {y'} = -2 \, {y} \hspace{1em} y( 1 ) = 2\]

Answer.

\[{y} = \frac{2}{t^{2}}\]


Example 45

F2: Separation of variables (ver. 45)

Find the solution to the given IVP.

\[{y} \sin\left(t\right) = {y'} \hspace{1em} y( 0 ) = e^{\left(-1\right)}\]

Answer.

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


Example 46

F2: Separation of variables (ver. 46)

Find the solution to the given IVP.

\[-3 \, {y} = t {y'} \hspace{1em} y( 2 ) = -\frac{1}{8}\]

Answer.

\[{y} = -\frac{1}{t^{3}}\]


Example 47

F2: Separation of variables (ver. 47)

Find the solution to the given IVP.

\[{y'} = {y} \sin\left(t\right) \hspace{1em} y( 0 ) = 3 \, e^{\left(-1\right)}\]

Answer.

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


Example 48

F2: Separation of variables (ver. 48)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = 0 \hspace{1em} y( 4 ) = -192\]

Answer.

\[{y} = -3 \, t^{3}\]


Example 49

F2: Separation of variables (ver. 49)

Find the solution to the given IVP.

\[t {y'} + 2 \, {y} = 0 \hspace{1em} y( -2 ) = 1\]

Answer.

\[{y} = \frac{4}{t^{2}}\]


Example 50

F2: Separation of variables (ver. 50)

Find the solution to the given IVP.

\[{\left(2 \, t + 1\right)} {y} - {y'} = 0 \hspace{1em} y( 0 ) = 4 \, e^{\left(-1\right)}\]

Answer.

\[{y} = 4 \, e^{\left(t^{2} + t - 1\right)}\]