F2: Separation of variables (ver. 1)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 2)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 3)

Find the solution to the given IVP.

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

${y} = 4 \, t^{3}$

F2: Separation of variables (ver. 4)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 5)

Find the solution to the given IVP.

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

${y} = -\frac{2}{t^{3}}$

F2: Separation of variables (ver. 6)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 7)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 8)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 9)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 10)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 11)

Find the solution to the given IVP.

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

${y} = -\frac{1}{t^{2}}$

F2: Separation of variables (ver. 12)

Find the solution to the given IVP.

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

${y} = -\frac{4}{t^{3}}$

F2: Separation of variables (ver. 13)

Find the solution to the given IVP.

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

${y} = -\frac{2}{t^{3}}$

F2: Separation of variables (ver. 14)

Find the solution to the given IVP.

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

${y} = -3 \, t^{2}$

F2: Separation of variables (ver. 15)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 16)

Find the solution to the given IVP.

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

${y} = -\frac{4}{t^{3}}$

F2: Separation of variables (ver. 17)

Find the solution to the given IVP.

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

${y} = -\frac{3}{t^{2}}$

F2: Separation of variables (ver. 18)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 19)

Find the solution to the given IVP.

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

${y} = 2 \, t^{3}$

F2: Separation of variables (ver. 20)

Find the solution to the given IVP.

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

${y} = -t^{3}$

F2: Separation of variables (ver. 21)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 22)

Find the solution to the given IVP.

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

${y} = \frac{4}{t^{2}}$

F2: Separation of variables (ver. 23)

Find the solution to the given IVP.

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

${y} = \frac{1}{t^{2}}$

F2: Separation of variables (ver. 24)

Find the solution to the given IVP.

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

${y} = -\frac{2}{t^{3}}$

F2: Separation of variables (ver. 25)

Find the solution to the given IVP.

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

${y} = -4 \, t^{3}$

F2: Separation of variables (ver. 26)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 27)

Find the solution to the given IVP.

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

${y} = -3 \, t^{3}$

F2: Separation of variables (ver. 28)

Find the solution to the given IVP.

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

${y} = 4 \, t^{3}$

F2: Separation of variables (ver. 29)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 30)

Find the solution to the given IVP.

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

${y} = \frac{1}{t^{3}}$

F2: Separation of variables (ver. 31)

Find the solution to the given IVP.

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

${y} = -2 \, t^{3}$

F2: Separation of variables (ver. 32)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 33)

Find the solution to the given IVP.

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

${y} = 3 \, t^{2}$

F2: Separation of variables (ver. 34)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 35)

Find the solution to the given IVP.

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

${y} = 4 \, t^{3}$

F2: Separation of variables (ver. 36)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 37)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 38)

Find the solution to the given IVP.

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

${y} = -\frac{4}{t^{2}}$

F2: Separation of variables (ver. 39)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 40)

Find the solution to the given IVP.

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

${y} = -\frac{3}{t^{2}}$

F2: Separation of variables (ver. 41)

Find the solution to the given IVP.

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

${y} = 4 \, t^{3}$

F2: Separation of variables (ver. 42)

Find the solution to the given IVP.

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

${y} = -2 \, t^{3}$

F2: Separation of variables (ver. 43)

Find the solution to the given IVP.

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

${y} = \frac{4}{t^{3}}$

F2: Separation of variables (ver. 44)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 45)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 46)

Find the solution to the given IVP.

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

${y} = -\frac{1}{t^{3}}$

F2: Separation of variables (ver. 47)

Find the solution to the given IVP.

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

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

F2: Separation of variables (ver. 48)

Find the solution to the given IVP.

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

${y} = -3 \, t^{3}$

F2: Separation of variables (ver. 49)

Find the solution to the given IVP.

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

${y} = \frac{4}{t^{2}}$

F2: Separation of variables (ver. 50)

Find the solution to the given IVP.

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

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