0066 points Previous Answers ZillDifEQModAp10 82007 Find th

0/0.66 points | Previous Answers ZillDifEQModAp10 8.2.007. Find the general solution of the given system. dt dt dz- ( C[1.00] ,[5,18,4]er+C3102lex Need Help? Read itTaik to a Tutor Submit Answer Save ProgressPractice Another Version

Solution

dy/dt=5y

INtegrating gives

y=Ae^{5t}

dz/dt=y-z

HEnce, dz/dt+z=Ae^{5t}

Integrating factor is e^t

Multiplying gives

(dz/dt+z)e^t=Ae^{6t}

(ze^t)\'=Ae^{6t}

Integrating gives

ze^t=Ae^{6t}/6+B

z=Ae^{5t}/6+Be^{-t}

dx/dt=x+Ae^{5t}-Ae^{5t}/6-Be^{-t}

dx/dt-x=5Ae^{5t}/6-Be^{-t}

Integrating factor is e^{-t}

Multiplying gives

(xe^{-t})\'=5Ae^{4t}/6-Be^{-2t}

INtegrating gives

xe^{-t}=5Ae^{4t}/24+Be^{-2t}/2+C

x=5Ae^{5t}/24+Be^{-t}/2+Ce^t

Let, C=C_1

A/24=C_2

A=24 C_2

C_3=B/2

B=2C_3

So,

<x,y,z>=<5C_2e^{5t}+C_3e^{-t}+C_1e^t,24C_2e^{5t},4C_2e^{5t}+2C_3e^{-t}>

            =C_1<1,0,0>e^t+C_2<5,24,4>e^{5t}+C_3<1,0,2>e^{-t}