(5).Solve the initial value problem for x x\'=3x+y y\'=-2x x(0)=1,y(0)=1 (a).x=-

Question

(5).Solve the initial value problem for x
x'=3x+y
y'=-2x
x(0)=1,y(0)=1
(a).x=-2e^t+3e^-t
(b).x=e^t-3e^3t
(c).x=e^t-3e^2t
(d).x=-2e^t+3e^2t

(6). Solve the initial value problem for y
x'=x+3y
y'=2x+2y
x(0)=1,y(0)=-1
(a). y=1/5e^4t+4/5e^-t
(b). y=-1/5e^4t-4/5e^-t
(c). y=-1/2e^t-e^2t
(d). y=e^4t+2/3e^t

(7). Solve the initial value problem for x
x'=-x+3y
y'=-2x+4y
x(0)=1,y(0)=-1
(a). x=e^t+3e^-t
(b). x=e^3t-2e^-t
(c). x=-1/2e^-t-e^2t
(d). x=e^2t

(8). Solve the initial value problem for y
x'=x+2y
y'=2x+y
x(0)=2,y(0)=-1
(a). y=e^4t+3e^-t
(b). y=e^3t-2e^-t
(c). y=-3/2e^-t+1/2e^3t
(d). y=e^-t+2e^3t

(9). Solve the initial value problem for y
x'=6x+2y
y'=2x+3y
x(0)=1,y(0)=1
(a). y=1/5e^4t+3e^-7t
(b). y=2/5e^2t+3/5e^7t
(c). y=-3/4e^-t+1/3e^3t
(d). y=e^-2t+2e^7t

Explanation / Answer

(5).Solve the initial value problem for x
x'=3x+y
y'=-2x
x(0)=1,y(0)=1
(a).x=-2e^t+3e^-t
(b).x=e^t-3e^3t
(c).x=e^t-3e^2t
(d).x=-2e^t+3e^2t     ANSWER   
solution:

x'=3x+y ==> x'' =3x' +y' ==> x''=3x' -2x ==> x''-3x'+2x=0
y'=-2x   

x''-3x'+2x=0

m^2-3m+2=

(m-3)(m-1)=0 ==> m=3 or m=1 ==>   x(t) = A e^t+ B e^2t
                                                                                x(0)=1, ==>   1= A+B

x'=3x+y ==>    y(t) = x' - 3x = A e^t+ 2 B e^2t - 3(A e^t+ B e^2t) = -2 A e^t - B e^2t

                 y(t) = -2 A e^t - B e^2t

                y(0)=1 ==> -2A -B=1

       1= A+B

      -2A -B=1    ==> -A= 2 ==> A+B=1==> -2 +B=1==> B=3

x(t) =    A e^t+ B e^2t   =   -2 e^t+ 3 e^2t    ANSWER

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