Question

For each DE in this set, classify its order = the highest number of derivatives of the unknown

function, and type: autonomous vs nonautonomous, linear vs nonlinear. If linear, say whether

it is homogeneous or nonhomogeneous, and whether it has constant coecients or nonconstant

coecients.

Answer 1

$x\left ( t \right )=C1e^{5t}+2C2e^{-t}$ ,    $y\left ( t \right )=C1e^{5t}-C2e^{-t}$

$\frac{dx}{dt}=x+4y,$         $\frac{dy}{dt}= 2x+3y$

$\frac{dx}{dt}=5C1e^{5t}-2C2e^{-t}= x+4y$

$x+4y= 5C1e^{5t}-2C2e^{-t} .........\left ( 1 \right )$

$\frac{dy}{dt}=5C1e^{5t}+C2e^{-t}= 2x+3y$

$2x+3y= 5C1e^{5t}+C2e^{-t} .............\left ( 2 \right )$

substarcting equation 2 from 1 we get

$x-y = 3C2e^{-t} ..................\left ( 3 \right )$

substarcting equation 1 from 2 we get

$x+4y-2x-3y=-2C2e^{-t}-C2e^{-t}$

$x+y= -3C2e^{-t} ...........................\left ( 4 \right )$

on adding equation 3 and 4 we get

we have 2x= 0 or x=0

from equation 4 we have

          

    and x=0

B) $x\left ( t=0 \right ) =0,$     $y\left ( t=0 \right ) =3$

$x\left ( t=0 \right )=C1e^{0}+2C2e^{0} =0$

$C1+2C2= 0 ..................\left ( 5 \right )$

$y\left ( t=0 \right )=C1e^{0}-C2e^{0} =3$

$C1-C2 =3 .................\left ( 6 \right )$

substarcting equation 5 from 6 we get

C1+2C2-C1+C2 = -3

3C2=-3,    C2=-1

from equation 5       C1+2C2=0

                C1-2=0,    C1=2.

therefore the solution is C1=2,    C2=-1