F3 - Techniques for linear IVPs


Example 1 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 2 πŸ”—

Find the solution to the given IVP.

\[ 0 = -t {y'} - 5 \, {y} + \frac{6}{t^{3}} \hspace{2em} y( -1 )= -1 \]

Answer:

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


Example 3 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 4 πŸ”—

Find the solution to the given IVP.

\[ {y'} t + 4 \, {y} + \frac{15}{t} = 0 \hspace{2em} y( -1 )= 1 \]

Answer:

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


Example 5 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 6 πŸ”—

Find the solution to the given IVP.

\[ 5 \, {y} - \frac{1}{t^{4}} = -{y'} t \hspace{2em} y( 1 )= 3 \]

Answer:

\[ {y} = \frac{1}{t^{4}} + \frac{2}{t^{5}} \]


Example 7 πŸ”—

Find the solution to the given IVP.

\[ -28 \, t^{3} - 4 \, {y} = {y'} t \hspace{2em} y( 1 )= -8 \]

Answer:

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


Example 8 πŸ”—

Find the solution to the given IVP.

\[ -3 \, {y} = 14 \, t^{4} + {y'} t \hspace{2em} y( 1 )= -6 \]

Answer:

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


Example 9 πŸ”—

Find the solution to the given IVP.

\[ -4 \, {y} = {y'} t - \frac{5}{t^{3}} \hspace{2em} y( 1 )= 2 \]

Answer:

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


Example 10 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 11 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 12 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 13 πŸ”—

Find the solution to the given IVP.

\[ 32 \, t^{4} = -{y'} t - 4 \, {y} \hspace{2em} y( -1 )= -3 \]

Answer:

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


Example 14 πŸ”—

Find the solution to the given IVP.

\[ 0 = -t {y'} + 3 \, {y} + \frac{35}{t^{4}} \hspace{2em} y( 1 )= -2 \]

Answer:

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


Example 15 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 16 πŸ”—

Find the solution to the given IVP.

\[ -{y'} t + 2 \, {y} - \frac{20}{t^{2}} = 0 \hspace{2em} y( 1 )= 7 \]

Answer:

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


Example 17 πŸ”—

Find the solution to the given IVP.

\[ -{y'} t - \frac{15}{t^{2}} = -3 \, {y} \hspace{2em} y( -1 )= 0 \]

Answer:

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


Example 18 πŸ”—

Find the solution to the given IVP.

\[ 0 = -{y'} t + 4 \, {y} - \frac{7}{t^{3}} \hspace{2em} y( 1 )= -1 \]

Answer:

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


Example 19 πŸ”—

Find the solution to the given IVP.

\[ -t {y'} + \frac{6}{t^{4}} = 2 \, {y} \hspace{2em} y( -1 )= -6 \]

Answer:

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


Example 20 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 21 πŸ”—

Find the solution to the given IVP.

\[ -\frac{15}{t^{2}} = -t {y'} - 5 \, {y} \hspace{2em} y( -1 )= 8 \]

Answer:

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


Example 22 πŸ”—

Find the solution to the given IVP.

\[ {y'} t + \frac{25}{t} = 4 \, {y} \hspace{2em} y( 1 )= 9 \]

Answer:

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


Example 23 πŸ”—

Find the solution to the given IVP.

\[ 0 = -{y'} t + 5 \, {y} - \frac{12}{t} \hspace{2em} y( -1 )= -6 \]

Answer:

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


Example 24 πŸ”—

Find the solution to the given IVP.

\[ 5 \, {y} = {y'} t - \frac{8}{t^{3}} \hspace{2em} y( -1 )= 2 \]

Answer:

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


Example 25 πŸ”—

Find the solution to the given IVP.

\[ -5 \, {y} + \frac{12}{t} = -t {y'} \hspace{2em} y( -1 )= 0 \]

Answer:

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


Example 26 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 27 πŸ”—

Find the solution to the given IVP.

\[ {y'} t - 3 \, {y} - \frac{35}{t^{4}} = 0 \hspace{2em} y( 1 )= -4 \]

Answer:

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


Example 28 πŸ”—

Find the solution to the given IVP.

\[ -{y'} t + 2 \, {y} - \frac{30}{t^{4}} = 0 \hspace{2em} y( -1 )= 7 \]

Answer:

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


Example 29 πŸ”—

Find the solution to the given IVP.

\[ 2 \, {y} = -8 \, t^{4} + {y'} t \hspace{2em} y( 1 )= 7 \]

Answer:

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


Example 30 πŸ”—

Find the solution to the given IVP.

\[ -40 \, t^{3} - 5 \, {y} = t {y'} \hspace{2em} y( -1 )= 2 \]

Answer:

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


Example 31 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 32 πŸ”—

Find the solution to the given IVP.

\[ -4 \, {y} - \frac{24}{t^{2}} = -{y'} t \hspace{2em} y( 1 )= -5 \]

Answer:

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


Example 33 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 34 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 35 πŸ”—

Find the solution to the given IVP.

\[ t {y'} - 5 \, {y} = \frac{36}{t^{4}} \hspace{2em} y( 1 )= -8 \]

Answer:

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


Example 36 πŸ”—

Find the solution to the given IVP.

\[ {y'} t + 10 \, t^{2} = 4 \, {y} \hspace{2em} y( 1 )= 1 \]

Answer:

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


Example 37 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 38 πŸ”—

Find the solution to the given IVP.

\[ -4 \, {y} = -14 \, t^{3} + {y'} t \hspace{2em} y( 1 )= 1 \]

Answer:

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


Example 39 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 40 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 41 πŸ”—

Find the solution to the given IVP.

\[ t {y'} - \frac{8}{t^{4}} = -2 \, {y} \hspace{2em} y( -1 )= -7 \]

Answer:

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


Example 42 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 43 πŸ”—

Find the solution to the given IVP.

\[ \frac{7}{t^{3}} = t {y'} - 4 \, {y} \hspace{2em} y( 1 )= 2 \]

Answer:

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


Example 44 πŸ”—

Find the solution to the given IVP.

\[ -t {y'} + 4 \, {y} + \frac{8}{t^{4}} = 0 \hspace{2em} y( -1 )= 2 \]

Answer:

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


Example 45 πŸ”—

Find the solution to the given IVP.

\[ -2 \, {y} = t {y'} + 12 \, t \hspace{2em} y( -1 )= 8 \]

Answer:

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


Example 46 πŸ”—

Find the solution to the given IVP.

\[ 5 \, {y} - \frac{6}{t^{3}} = -{y'} t \hspace{2em} y( -1 )= -7 \]

Answer:

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


Example 47 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 48 πŸ”—

Find the solution to the given IVP.

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

Answer:

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


Example 49 πŸ”—

Find the solution to the given IVP.

\[ 0 = 4 \, t^{4} - {y'} t + 5 \, {y} \hspace{2em} y( 1 )= -8 \]

Answer:

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


Example 50 πŸ”—

Find the solution to the given IVP.

\[ -{y'} t + 5 \, {y} = \frac{12}{t} \hspace{2em} y( 1 )= 3 \]

Answer:

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