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## anonymous 3 years ago 3d2y/dt2-dy/dt +2y=e-3t Differential Equation

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1. anonymous

$\frac{d^2 y}{dt^2}-\frac{1}{3} \frac{dy}{dt}+\frac{2 y}{3}=\frac{e^{-3t}}{3}$ Solve the homogenous part (=0). Then try a solution of the form: $y_p(t)=Ae^{-3t}$ And differentiate and plug in. Be careful that your homogenous solution doesn't have a root of 3 or else you need something like: $y_p(t)=Ate^{-3t}$

2. anonymous

the characteristic root of the homogeneous DE are complex numbers, different from the characteristic root of the RHS. writing the LHS as $$f(D)y$$ you can determine right away the particular solution $y_p=\frac{e^{-3t}}{f(-3)}$

3. anonymous

minor correction:$\Huge y_p=\frac{\frac{e^{-3t}}{3}}{f(-3)}$

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