## Dugee Group Title Does anyone know how to solve this? Solve the differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0. y'' + 2y' − 8y = 5e^(−3x) − e^(−x) I get y = c1e^(-4x)+c2e^(2x)-e^(-3x)+e^(-x)/9 but idk how to carry on with the initial condition part. one year ago one year ago

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1. sirm3d

$\Large {y=c_1e^{-4x}+c_2e^{2x}-e^{-3x}+e^{-x}/9\\1=c_1+c_2-1+1/9}$ $\Large {y'=-4c_1e^{-4x}+2c_2e^{2x}+3e^{-3x}-e^{-x}/9\\0=-4c_1+2c_2+3-1/9}$ solve the unknowns $$\large c_1,\;c_2$$

2. Dugee

Thanks. I was able to solve the unknowns. $c_{1} = \frac{ 121 }{ 54 } , c_{2} = -\frac{ 7 }{54 }$ Is that what you got as well?