## F3 - Techniques for linear IVPs

#### Example 1 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

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 π

Find the solution to the given IVP.

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

Answer:

${y} = t^{5} + \frac{2}{t}$