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(Solved): SOLVE the following system of differential equations: x=6x3y+8ety=y+2x+4etx(0)=1y(0)= ...



SOLVE the following system of differential equations:
\[
\begin{array}{l}
\dot{x}=6 x-3 y+8 e^{t} \\
\dot{y}=y+2 x+4 e^{t} \\

SOLVE the following system of differential equations:


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(A): Given the differential equations:

x' = 6x - 3y + 8et
y' = y + 2x + 4et
The given conditions are:

x(0) = -1
y(0) = 0

To solve the system, we can use the method of integrating factors.
Then, the integrating factor for the first equation is given by:
  


note that the differentiation had done to the above equation.

Multiply both sides of the first equation by   :


that means we have to substitute the equation in the first (x') differential equation.


e6tx' = 6e6tx - 3e6ty + 8et*e6t

e6tx' = 6e6tx - 3e6ty + 8e7t




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