## X1 - Linear ODE systems

#### Example 1 π

Find the solution to the given system of IVPs.

$2 \, {x} = -9 \, {y} - {x'} \hspace{2em}x(0)= -8$

$-7 \, {y} - 4 \, {x} = {y'} \hspace{2em}x(0)= 5$

$x= -9 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}$

$y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-11 \, t\right)}$

#### Example 2 π

Find the solution to the given system of IVPs.

$0 = -5 \, {x} + 2 \, {y} + {x'} \hspace{2em}x(0)= -3$

$-2 \, {x} = {y'} - 8 \, {y} \hspace{2em}x(0)= 1$

$x= -e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}$

$y= 2 \, e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}$

#### Example 3 π

Find the solution to the given system of IVPs.

$-{x'} - 4 \, {y} + 5 \, {x} = 0 \hspace{2em}x(0)= -3$

${x} = 8 \, {y} - {y'} \hspace{2em}x(0)= -2$

$x= e^{\left(9 \, t\right)} - 4 \, e^{\left(4 \, t\right)}$

$y= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}$

#### Example 4 π

Find the solution to the given system of IVPs.

$-2 \, {y} = -2 \, {x} + {x'} \hspace{2em}x(0)= 3$

$0 = -2 \, {x} + 5 \, {y} - {y'} \hspace{2em}x(0)= -1$

$x= e^{\left(6 \, t\right)} + 2 \, e^{t}$

$y= -2 \, e^{\left(6 \, t\right)} + e^{t}$

#### Example 5 π

Find the solution to the given system of IVPs.

${x'} - 2 \, {x} - 4 \, {y} = 0 \hspace{2em}x(0)= -5$

$5 \, {y} = -{x} + {y'} \hspace{2em}x(0)= 0$

$x= -e^{\left(6 \, t\right)} - 4 \, e^{t}$

$y= -e^{\left(6 \, t\right)} + e^{t}$

#### Example 6 π

Find the solution to the given system of IVPs.

$2 \, {y} - 4 \, {x} = -{x'} \hspace{2em}x(0)= 1$

$7 \, {y} = 2 \, {x} + {y'} \hspace{2em}x(0)= 3$

$x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}$

$y= 2 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}$

#### Example 7 π

Find the solution to the given system of IVPs.

${x'} = 3 \, {x} - 4 \, {y} \hspace{2em}x(0)= 5$

$8 \, {y} - {y'} = -{x} \hspace{2em}x(0)= -2$

$x= e^{\left(7 \, t\right)} + 4 \, e^{\left(4 \, t\right)}$

$y= -e^{\left(7 \, t\right)} - e^{\left(4 \, t\right)}$

#### Example 8 π

Find the solution to the given system of IVPs.

$-{y} + {x} = -{x'} \hspace{2em}x(0)= 0$

$0 = 6 \, {y} + 4 \, {x} + {y'} \hspace{2em}x(0)= -3$

$x= -e^{\left(-2 \, t\right)} + e^{\left(-5 \, t\right)}$

$y= e^{\left(-2 \, t\right)} - 4 \, e^{\left(-5 \, t\right)}$

#### Example 9 π

Find the solution to the given system of IVPs.

$-9 \, {y} = {x} - {x'} \hspace{2em}x(0)= 10$

$4 \, {x} - {y'} - 4 \, {y} = 0 \hspace{2em}x(0)= 3$

$x= 9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}$

$y= 4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}$

#### Example 10 π

Find the solution to the given system of IVPs.

$-4 \, {y} = -2 \, {x} + {x'} \hspace{2em}x(0)= 5$

${x} = 5 \, {y} - {y'} \hspace{2em}x(0)= 0$

$x= e^{\left(6 \, t\right)} + 4 \, e^{t}$

$y= -e^{\left(6 \, t\right)} + e^{t}$

#### Example 11 π

Find the solution to the given system of IVPs.

${x'} + 2 \, {x} + {y} = 0 \hspace{2em}x(0)= 0$

$-7 \, {y} = -4 \, {x} + {y'} \hspace{2em}x(0)= 3$

$x= -e^{\left(-3 \, t\right)} + e^{\left(-6 \, t\right)}$

$y= -e^{\left(-3 \, t\right)} + 4 \, e^{\left(-6 \, t\right)}$

#### Example 12 π

Find the solution to the given system of IVPs.

${x'} = -2 \, {x} + {y} \hspace{2em}x(0)= 0$

$-4 \, {x} + 5 \, {y} = -{y'} \hspace{2em}x(0)= -5$

$x= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}$

$y= -e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}$

#### Example 13 π

Find the solution to the given system of IVPs.

${x'} = {y} - {x} \hspace{2em}x(0)= 0$

$6 \, {y} + {y'} + 4 \, {x} = 0 \hspace{2em}x(0)= 3$

$x= e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}$

$y= -e^{\left(-2 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}$

#### Example 14 π

Find the solution to the given system of IVPs.

${x'} - {y} = 2 \, {x} \hspace{2em}x(0)= -2$

${y'} + 3 \, {y} + 4 \, {x} = 0 \hspace{2em}x(0)= 5$

$x= -e^{\left(-2 \, t\right)} - e^{t}$

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

#### Example 15 π

Find the solution to the given system of IVPs.

$0 = -4 \, {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5$

$9 \, {y} = -{x} + {y'} \hspace{2em}x(0)= -2$

$x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}$

$y= -e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}$

#### Example 16 π

Find the solution to the given system of IVPs.

$-{x'} = -3 \, {y} + 4 \, {x} \hspace{2em}x(0)= -4$

${y'} - {y} = 12 \, {x} \hspace{2em}x(0)= 1$

$x= -e^{\left(5 \, t\right)} - 3 \, e^{\left(-8 \, t\right)}$

$y= -3 \, e^{\left(5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}$

#### Example 17 π

Find the solution to the given system of IVPs.

${x'} = {x} + 9 \, {y} \hspace{2em}x(0)= -8$

$0 = 4 \, {x} - {y'} - 4 \, {y} \hspace{2em}x(0)= -5$

$x= -9 \, e^{\left(5 \, t\right)} + e^{\left(-8 \, t\right)}$

$y= -4 \, e^{\left(5 \, t\right)} - e^{\left(-8 \, t\right)}$

#### Example 18 π

Find the solution to the given system of IVPs.

$-{x} + {y} - {x'} = 0 \hspace{2em}x(0)= -2$

$-2 \, {y} + {y'} = 4 \, {x} \hspace{2em}x(0)= -3$

$x= -e^{\left(3 \, t\right)} - e^{\left(-2 \, t\right)}$

$y= -4 \, e^{\left(3 \, t\right)} + e^{\left(-2 \, t\right)}$

#### Example 19 π

Find the solution to the given system of IVPs.

$-9 \, {y} = {x'} + 3 \, {x} \hspace{2em}x(0)= 10$

$-{y'} - 4 \, {x} = 8 \, {y} \hspace{2em}x(0)= -3$

$x= e^{\left(-12 \, t\right)} + 9 \, e^{t}$

$y= e^{\left(-12 \, t\right)} - 4 \, e^{t}$

#### Example 20 π

Find the solution to the given system of IVPs.

$-{y} = -{x'} - 2 \, {x} \hspace{2em}x(0)= 2$

$-4 \, {x} = -{y'} - 5 \, {y} \hspace{2em}x(0)= -3$

$x= e^{\left(-t\right)} + e^{\left(-6 \, t\right)}$

$y= e^{\left(-t\right)} - 4 \, e^{\left(-6 \, t\right)}$

#### Example 21 π

Find the solution to the given system of IVPs.

$-4 \, {x} + {y} + {x'} = 0 \hspace{2em}x(0)= 0$

$-4 \, {x} - {y'} = -7 \, {y} \hspace{2em}x(0)= 5$

$x= -e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}$

$y= 4 \, e^{\left(8 \, t\right)} + e^{\left(3 \, t\right)}$

#### Example 22 π

Find the solution to the given system of IVPs.

$3 \, {y} - 3 \, {x} = {x'} \hspace{2em}x(0)= -4$

$8 \, {y} + {y'} - 12 \, {x} = 0 \hspace{2em}x(0)= -1$

$x= -e^{\left(-12 \, t\right)} - 3 \, e^{t}$

$y= 3 \, e^{\left(-12 \, t\right)} - 4 \, e^{t}$

#### Example 23 π

Find the solution to the given system of IVPs.

$2 \, {x} - {x'} = 3 \, {y} \hspace{2em}x(0)= -4$

$7 \, {y} = 12 \, {x} + {y'} \hspace{2em}x(0)= -1$

$x= -e^{\left(11 \, t\right)} - 3 \, e^{\left(-2 \, t\right)}$

$y= 3 \, e^{\left(11 \, t\right)} - 4 \, e^{\left(-2 \, t\right)}$

#### Example 24 π

Find the solution to the given system of IVPs.

${y} = 3 \, {x} + {x'} \hspace{2em}x(0)= 2$

$2 \, {y} = 4 \, {x} + {y'} \hspace{2em}x(0)= 5$

$x= e^{\left(-2 \, t\right)} + e^{t}$

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

#### Example 25 π

Find the solution to the given system of IVPs.

$0 = -{x} - 2 \, {y} + {x'} \hspace{2em}x(0)= 1$

$2 \, {y} = -{y'} + 2 \, {x} \hspace{2em}x(0)= 3$

$x= 2 \, e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}$

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

#### Example 26 π

Find the solution to the given system of IVPs.

$-4 \, {x} = -{x'} - 12 \, {y} \hspace{2em}x(0)= 1$

${y} = -{y'} - 3 \, {x} \hspace{2em}x(0)= 4$

$x= -3 \, e^{\left(8 \, t\right)} + 4 \, e^{\left(-5 \, t\right)}$

$y= e^{\left(8 \, t\right)} + 3 \, e^{\left(-5 \, t\right)}$

#### Example 27 π

Find the solution to the given system of IVPs.

$0 = {x} - {y} - {x'} \hspace{2em}x(0)= -2$

${y'} = -4 \, {x} - 2 \, {y} \hspace{2em}x(0)= -3$

$x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}$

$y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}$

#### Example 28 π

Find the solution to the given system of IVPs.

$-{x'} - {x} + 2 \, {y} = 0 \hspace{2em}x(0)= -3$

${y'} = -2 \, {x} - 6 \, {y} \hspace{2em}x(0)= 3$

$x= -2 \, e^{\left(-2 \, t\right)} - e^{\left(-5 \, t\right)}$

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

#### Example 29 π

Find the solution to the given system of IVPs.

$0 = -4 \, {x} + {x'} + 4 \, {y} \hspace{2em}x(0)= 5$

$9 \, {y} = -{x} + {y'} \hspace{2em}x(0)= -2$

$x= e^{\left(8 \, t\right)} + 4 \, e^{\left(5 \, t\right)}$

$y= -e^{\left(8 \, t\right)} - e^{\left(5 \, t\right)}$

#### Example 30 π

Find the solution to the given system of IVPs.

${y} - 5 \, {x} = -{x'} \hspace{2em}x(0)= 0$

$-2 \, {y} + {y'} = -4 \, {x} \hspace{2em}x(0)= 5$

$x= -e^{\left(6 \, t\right)} + e^{t}$

$y= e^{\left(6 \, t\right)} + 4 \, e^{t}$

#### Example 31 π

Find the solution to the given system of IVPs.

${x} + 4 \, {y} = {x'} \hspace{2em}x(0)= -3$

${x} = {y'} + 2 \, {y} \hspace{2em}x(0)= -2$

$x= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}$

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

#### Example 32 π

Find the solution to the given system of IVPs.

$0 = -9 \, {y} - {x'} + {x} \hspace{2em}x(0)= -8$

$-6 \, {y} + 4 \, {x} = -{y'} \hspace{2em}x(0)= -5$

$x= e^{\left(10 \, t\right)} - 9 \, e^{\left(-3 \, t\right)}$

$y= -e^{\left(10 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}$

#### Example 33 π

Find the solution to the given system of IVPs.

$-{x'} = -4 \, {y} + 2 \, {x} \hspace{2em}x(0)= -3$

$-3 \, {y} + {x} = -{y'} \hspace{2em}x(0)= 0$

$x= e^{\left(2 \, t\right)} - 4 \, e^{\left(-t\right)}$

$y= e^{\left(2 \, t\right)} - e^{\left(-t\right)}$

#### Example 34 π

Find the solution to the given system of IVPs.

$-4 \, {y} - {x'} - 2 \, {x} = 0 \hspace{2em}x(0)= 5$

${x} = -{y'} - 5 \, {y} \hspace{2em}x(0)= 0$

$x= 4 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}$

$y= -e^{\left(-t\right)} + e^{\left(-6 \, t\right)}$

#### Example 35 π

Find the solution to the given system of IVPs.

$0 = {x'} + 2 \, {y} + 2 \, {x} \hspace{2em}x(0)= -1$

$5 \, {y} + 2 \, {x} + {y'} = 0 \hspace{2em}x(0)= 3$

$x= -2 \, e^{\left(-t\right)} + e^{\left(-6 \, t\right)}$

$y= e^{\left(-t\right)} + 2 \, e^{\left(-6 \, t\right)}$

#### Example 36 π

Find the solution to the given system of IVPs.

$-{x'} = -{y} - 5 \, {x} \hspace{2em}x(0)= -2$

$-{y'} = -8 \, {y} - 4 \, {x} \hspace{2em}x(0)= -3$

$x= -e^{\left(9 \, t\right)} - e^{\left(4 \, t\right)}$

$y= -4 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}$

#### Example 37 π

Find the solution to the given system of IVPs.

$-{x'} - 2 \, {x} + {y} = 0 \hspace{2em}x(0)= -2$

$4 \, {x} = {y'} + 5 \, {y} \hspace{2em}x(0)= 3$

$x= -e^{\left(-t\right)} - e^{\left(-6 \, t\right)}$

$y= -e^{\left(-t\right)} + 4 \, e^{\left(-6 \, t\right)}$

#### Example 38 π

Find the solution to the given system of IVPs.

$6 \, {y} = -{x} - {x'} \hspace{2em}x(0)= -5$

$-6 \, {x} = {y'} + 6 \, {y} \hspace{2em}x(0)= -1$

$x= -3 \, e^{\left(3 \, t\right)} - 2 \, e^{\left(-10 \, t\right)}$

$y= 2 \, e^{\left(3 \, t\right)} - 3 \, e^{\left(-10 \, t\right)}$

#### Example 39 π

Find the solution to the given system of IVPs.

$2 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1$

$-{y'} + 2 \, {x} = -7 \, {y} \hspace{2em}x(0)= -3$

$x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}$

$y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}$

#### Example 40 π

Find the solution to the given system of IVPs.

$-4 \, {x} + {y} = -{x'} \hspace{2em}x(0)= -2$

$-7 \, {y} + {y'} = -4 \, {x} \hspace{2em}x(0)= 3$

$x= -e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}$

$y= 4 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}$

#### Example 41 π

Find the solution to the given system of IVPs.

$-{x'} = -{x} - 12 \, {y} \hspace{2em}x(0)= -1$

$3 \, {x} = 4 \, {y} + {y'} \hspace{2em}x(0)= 4$

$x= 3 \, e^{\left(5 \, t\right)} - 4 \, e^{\left(-8 \, t\right)}$

$y= e^{\left(5 \, t\right)} + 3 \, e^{\left(-8 \, t\right)}$

#### Example 42 π

Find the solution to the given system of IVPs.

$2 \, {y} = -5 \, {x} + {x'} \hspace{2em}x(0)= -1$

$8 \, {y} + 2 \, {x} - {y'} = 0 \hspace{2em}x(0)= 3$

$x= e^{\left(9 \, t\right)} - 2 \, e^{\left(4 \, t\right)}$

$y= 2 \, e^{\left(9 \, t\right)} + e^{\left(4 \, t\right)}$

#### Example 43 π

Find the solution to the given system of IVPs.

$2 \, {y} + 3 \, {x} = {x'} \hspace{2em}x(0)= 1$

$-8 \, {y} + 2 \, {x} = -{y'} \hspace{2em}x(0)= -1$

$x= -e^{\left(7 \, t\right)} + 2 \, e^{\left(4 \, t\right)}$

$y= -2 \, e^{\left(7 \, t\right)} + e^{\left(4 \, t\right)}$

#### Example 44 π

Find the solution to the given system of IVPs.

$-{y} + {x'} = -2 \, {x} \hspace{2em}x(0)= -2$

$0 = {y} + 4 \, {x} - {y'} \hspace{2em}x(0)= -3$

$x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}$

$y= -4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}$

#### Example 45 π

Find the solution to the given system of IVPs.

${y} - 2 \, {x} = {x'} \hspace{2em}x(0)= 0$

${y} + 4 \, {x} = {y'} \hspace{2em}x(0)= 5$

$x= e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}$

$y= 4 \, e^{\left(2 \, t\right)} + e^{\left(-3 \, t\right)}$

#### Example 46 π

Find the solution to the given system of IVPs.

$2 \, {y} = {x'} - 4 \, {x} \hspace{2em}x(0)= 1$

$2 \, {x} = -7 \, {y} + {y'} \hspace{2em}x(0)= -3$

$x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(3 \, t\right)}$

$y= -2 \, e^{\left(8 \, t\right)} - e^{\left(3 \, t\right)}$

#### Example 47 π

Find the solution to the given system of IVPs.

${y} - 4 \, {x} - {x'} = 0 \hspace{2em}x(0)= 0$

$-9 \, {y} - {y'} = 4 \, {x} \hspace{2em}x(0)= 3$

$x= e^{\left(-5 \, t\right)} - e^{\left(-8 \, t\right)}$

$y= -e^{\left(-5 \, t\right)} + 4 \, e^{\left(-8 \, t\right)}$

#### Example 48 π

Find the solution to the given system of IVPs.

$2 \, {y} - {x'} = -4 \, {x} \hspace{2em}x(0)= 1$

$-{y'} = -9 \, {y} + 2 \, {x} \hspace{2em}x(0)= -1$

$x= -e^{\left(8 \, t\right)} + 2 \, e^{\left(5 \, t\right)}$

$y= -2 \, e^{\left(8 \, t\right)} + e^{\left(5 \, t\right)}$

#### Example 49 π

Find the solution to the given system of IVPs.

$-{x'} - {x} = 4 \, {y} \hspace{2em}x(0)= 5$

$0 = 2 \, {y} - {x} - {y'} \hspace{2em}x(0)= 0$

$x= e^{\left(3 \, t\right)} + 4 \, e^{\left(-2 \, t\right)}$

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

#### Example 50 π

Find the solution to the given system of IVPs.

${x} - {y} = {x'} \hspace{2em}x(0)= -2$

$-4 \, {x} - 2 \, {y} - {y'} = 0 \hspace{2em}x(0)= -3$

$x= -e^{\left(2 \, t\right)} - e^{\left(-3 \, t\right)}$
$y= e^{\left(2 \, t\right)} - 4 \, e^{\left(-3 \, t\right)}$