F5: First order linear IVP


Example 1

F5: First order linear IVP (ver. 1)

Find the solution to the given IVP.

\[-\frac{24}{t} = -t {y'} + 5 \, {y} \hspace{1em} y( -1 ) = 3\]

Answer.

\[{y} = t^{5} - \frac{4}{t}\]


Example 2

F5: First order linear IVP (ver. 2)

Find the solution to the given IVP.

\[-\frac{4}{t^{4}} = -t {y'} - 5 \, {y} \hspace{1em} y( -1 ) = 6\]

Answer.

\[{y} = \frac{4}{t^{4}} - \frac{2}{t^{5}}\]


Example 3

F5: First order linear IVP (ver. 3)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} + 8 \, t \hspace{1em} y( 1 ) = 6\]

Answer.

\[{y} = 2 \, t^{3} + 4 \, t\]


Example 4

F5: First order linear IVP (ver. 4)

Find the solution to the given IVP.

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

Answer.

\[{y} = \frac{1}{t} + \frac{3}{t^{4}}\]


Example 5

F5: First order linear IVP (ver. 5)

Find the solution to the given IVP.

\[t {y'} - 2 \, {y} = \frac{8}{t^{2}} \hspace{1em} y( -1 ) = 0\]

Answer.

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


Example 6

F5: First order linear IVP (ver. 6)

Find the solution to the given IVP.

\[t {y'} + \frac{27}{t^{4}} = 5 \, {y} \hspace{1em} y( -1 ) = 1\]

Answer.

\[{y} = 2 \, t^{5} + \frac{3}{t^{4}}\]


Example 7

F5: First order linear IVP (ver. 7)

Find the solution to the given IVP.

\[6 \, t^{3} = -t {y'} + 5 \, {y} \hspace{1em} y( 1 ) = 7\]

Answer.

\[{y} = 4 \, t^{5} + 3 \, t^{3}\]


Example 8

F5: First order linear IVP (ver. 8)

Find the solution to the given IVP.

\[t {y'} - 5 \, {y} = -\frac{40}{t^{3}} \hspace{1em} y( 1 ) = 6\]

Answer.

\[{y} = t^{5} + \frac{5}{t^{3}}\]


Example 9

F5: First order linear IVP (ver. 9)

Find the solution to the given IVP.

\[t {y'} + 4 \, {y} = 24 \, t^{4} \hspace{1em} y( 1 ) = 1\]

Answer.

\[{y} = 3 \, t^{4} - \frac{2}{t^{4}}\]


Example 10

F5: First order linear IVP (ver. 10)

Find the solution to the given IVP.

\[7 \, t^{2} + t {y'} = -5 \, {y} \hspace{1em} y( -1 ) = -2\]

Answer.

\[{y} = -t^{2} + \frac{1}{t^{5}}\]


Example 11

F5: First order linear IVP (ver. 11)

Find the solution to the given IVP.

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

Answer.

\[{y} = -\frac{1}{t^{2}} - \frac{1}{t^{3}}\]


Example 12

F5: First order linear IVP (ver. 12)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = -5 \, t^{4} \hspace{1em} y( -1 ) = -1\]

Answer.

\[{y} = -5 \, t^{4} - 4 \, t^{3}\]


Example 13

F5: First order linear IVP (ver. 13)

Find the solution to the given IVP.

\[t {y'} - \frac{10}{t^{3}} = 2 \, {y} \hspace{1em} y( -1 ) = -1\]

Answer.

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


Example 14

F5: First order linear IVP (ver. 14)

Find the solution to the given IVP.

\[t {y'} + 2 \, {y} = -18 \, t^{4} \hspace{1em} y( 1 ) = -4\]

Answer.

\[{y} = -3 \, t^{4} - \frac{1}{t^{2}}\]


Example 15

F5: First order linear IVP (ver. 15)

Find the solution to the given IVP.

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

Answer.

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


Example 16

F5: First order linear IVP (ver. 16)

Find the solution to the given IVP.

\[t {y'} - 10 \, t = 3 \, {y} \hspace{1em} y( -1 ) = 9\]

Answer.

\[{y} = -4 \, t^{3} - 5 \, t\]


Example 17

F5: First order linear IVP (ver. 17)

Find the solution to the given IVP.

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

Answer.

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


Example 18

F5: First order linear IVP (ver. 18)

Find the solution to the given IVP.

\[-\frac{12}{t} = -t {y'} - 5 \, {y} \hspace{1em} y( 1 ) = 5\]

Answer.

\[{y} = \frac{3}{t} + \frac{2}{t^{5}}\]


Example 19

F5: First order linear IVP (ver. 19)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} + \frac{10}{t^{2}} \hspace{1em} y( 1 ) = -1\]

Answer.

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


Example 20

F5: First order linear IVP (ver. 20)

Find the solution to the given IVP.

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

Answer.

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


Example 21

F5: First order linear IVP (ver. 21)

Find the solution to the given IVP.

\[-12 \, t^{4} = -t {y'} - 2 \, {y} \hspace{1em} y( -1 ) = 5\]

Answer.

\[{y} = 2 \, t^{4} + \frac{3}{t^{2}}\]


Example 22

F5: First order linear IVP (ver. 22)

Find the solution to the given IVP.

\[-30 \, t^{3} = -t {y'} - 3 \, {y} \hspace{1em} y( -1 ) = -6\]

Answer.

\[{y} = 5 \, t^{3} + \frac{1}{t^{3}}\]


Example 23

F5: First order linear IVP (ver. 23)

Find the solution to the given IVP.

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

Answer.

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


Example 24

F5: First order linear IVP (ver. 24)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} - \frac{10}{t^{2}} \hspace{1em} y( -1 ) = -5\]

Answer.

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


Example 25

F5: First order linear IVP (ver. 25)

Find the solution to the given IVP.

\[t {y'} - \frac{20}{t} = 3 \, {y} \hspace{1em} y( 1 ) = -8\]

Answer.

\[{y} = -3 \, t^{3} - \frac{5}{t}\]


Example 26

F5: First order linear IVP (ver. 26)

Find the solution to the given IVP.

\[t {y'} + \frac{45}{t^{4}} = 5 \, {y} \hspace{1em} y( -1 ) = 2\]

Answer.

\[{y} = 3 \, t^{5} + \frac{5}{t^{4}}\]


Example 27

F5: First order linear IVP (ver. 27)

Find the solution to the given IVP.

\[t {y'} - 2 \, {y} = \frac{16}{t^{2}} \hspace{1em} y( 1 ) = -6\]

Answer.

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


Example 28

F5: First order linear IVP (ver. 28)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} + \frac{24}{t^{3}} \hspace{1em} y( 1 ) = 2\]

Answer.

\[{y} = -2 \, t^{3} + \frac{4}{t^{3}}\]


Example 29

F5: First order linear IVP (ver. 29)

Find the solution to the given IVP.

\[t {y'} - 5 \, {y} = \frac{24}{t^{3}} \hspace{1em} y( -1 ) = 2\]

Answer.

\[{y} = t^{5} - \frac{3}{t^{3}}\]


Example 30

F5: First order linear IVP (ver. 30)

Find the solution to the given IVP.

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

Answer.

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


Example 31

F5: First order linear IVP (ver. 31)

Find the solution to the given IVP.

\[2 \, {y} = t {y'} - \frac{12}{t^{2}} \hspace{1em} y( 1 ) = -2\]

Answer.

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


Example 32

F5: First order linear IVP (ver. 32)

Find the solution to the given IVP.

\[t {y'} - 4 \, {y} = -\frac{35}{t^{3}} \hspace{1em} y( 1 ) = 1\]

Answer.

\[{y} = -4 \, t^{4} + \frac{5}{t^{3}}\]


Example 33

F5: First order linear IVP (ver. 33)

Find the solution to the given IVP.

\[-24 \, t^{4} = -t {y'} - 2 \, {y} \hspace{1em} y( 1 ) = 0\]

Answer.

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


Example 34

F5: First order linear IVP (ver. 34)

Find the solution to the given IVP.

\[t {y'} - \frac{30}{t^{2}} = 4 \, {y} \hspace{1em} y( -1 ) = -3\]

Answer.

\[{y} = 2 \, t^{4} - \frac{5}{t^{2}}\]


Example 35

F5: First order linear IVP (ver. 35)

Find the solution to the given IVP.

\[3 \, {y} = t {y'} + \frac{35}{t^{4}} \hspace{1em} y( 1 ) = 9\]

Answer.

\[{y} = 4 \, t^{3} + \frac{5}{t^{4}}\]


Example 36

F5: First order linear IVP (ver. 36)

Find the solution to the given IVP.

\[t {y'} + 5 \, {y} = -\frac{1}{t^{4}} \hspace{1em} y( 1 ) = -2\]

Answer.

\[{y} = -\frac{1}{t^{4}} - \frac{1}{t^{5}}\]


Example 37

F5: First order linear IVP (ver. 37)

Find the solution to the given IVP.

\[t {y'} - 5 \, {y} = \frac{24}{t^{3}} \hspace{1em} y( 1 ) = -5\]

Answer.

\[{y} = -2 \, t^{5} - \frac{3}{t^{3}}\]


Example 38

F5: First order linear IVP (ver. 38)

Find the solution to the given IVP.

\[t {y'} + \frac{5}{t} = -2 \, {y} \hspace{1em} y( -1 ) = 2\]

Answer.

\[{y} = -\frac{5}{t} - \frac{3}{t^{2}}\]


Example 39

F5: First order linear IVP (ver. 39)

Find the solution to the given IVP.

\[-5 \, {y} = t {y'} - \frac{6}{t^{2}} \hspace{1em} y( 1 ) = 0\]

Answer.

\[{y} = \frac{2}{t^{2}} - \frac{2}{t^{5}}\]


Example 40

F5: First order linear IVP (ver. 40)

Find the solution to the given IVP.

\[t {y'} + 3 \, {y} = -20 \, t^{2} \hspace{1em} y( -1 ) = -5\]

Answer.

\[{y} = -4 \, t^{2} + \frac{1}{t^{3}}\]


Example 41

F5: First order linear IVP (ver. 41)

Find the solution to the given IVP.

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

Answer.

\[{y} = 5 \, t^{4} - t^{3}\]


Example 42

F5: First order linear IVP (ver. 42)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = \frac{8}{t} \hspace{1em} y( 1 ) = -5\]

Answer.

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


Example 43

F5: First order linear IVP (ver. 43)

Find the solution to the given IVP.

\[\frac{30}{t^{4}} = -t {y'} + 2 \, {y} \hspace{1em} y( -1 ) = 8\]

Answer.

\[{y} = 3 \, t^{2} + \frac{5}{t^{4}}\]


Example 44

F5: First order linear IVP (ver. 44)

Find the solution to the given IVP.

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

Answer.

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


Example 45

F5: First order linear IVP (ver. 45)

Find the solution to the given IVP.

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

Answer.

\[{y} = -3 \, t^{5} - 2 \, t^{4}\]


Example 46

F5: First order linear IVP (ver. 46)

Find the solution to the given IVP.

\[t {y'} - 3 \, {y} = \frac{7}{t^{4}} \hspace{1em} y( -1 ) = -3\]

Answer.

\[{y} = 2 \, t^{3} - \frac{1}{t^{4}}\]


Example 47

F5: First order linear IVP (ver. 47)

Find the solution to the given IVP.

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

Answer.

\[{y} = -\frac{2}{t^{4}} - \frac{2}{t^{5}}\]


Example 48

F5: First order linear IVP (ver. 48)

Find the solution to the given IVP.

\[2 \, {y} = t {y'} - \frac{16}{t^{2}} \hspace{1em} y( 1 ) = 0\]

Answer.

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


Example 49

F5: First order linear IVP (ver. 49)

Find the solution to the given IVP.

\[-25 \, t^{2} + t {y'} = -3 \, {y} \hspace{1em} y( -1 ) = 7\]

Answer.

\[{y} = 5 \, t^{2} - \frac{2}{t^{3}}\]


Example 50

F5: First order linear IVP (ver. 50)

Find the solution to the given IVP.

\[-4 \, {y} = 14 \, t^{3} + t {y'} \hspace{1em} y( 1 ) = -5\]

Answer.

\[{y} = -2 \, t^{3} - \frac{3}{t^{4}}\]