pls explain the concept of homogeneous differential equation

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pls explain the concept of homogeneous differential equation

Mathematics
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A homogenous differential equation is something in the form: \[y(t) \left[ \sum_{k=0}^{n} \left( f_n(t) \frac{d^n}{dt^n} \right) \right]=0\] Where f_n(t) is some arbitrary function multiplying the n^th differential operator acting on y(t). A homogenous equation is just one that is equal to zero. For example: \[y''(t)+y'(t)=0\] Or: \[\sin(t)y'(t)+y(t)\cos(3t)e^{2t}=0\]
how do u solve it
Well, if you have something like: \[y''+3y'+2=0\] You can rewrite it as a polynomial (as something to do with linear algebra and eigenvalues) \[\phi^2+3\phi+2=0 \implies (\phi+2)(\phi+1)=0 \implies \phi=-2,-1\] So the solution is: \[y(t)=c_1e^{-2t}+c_2e^{-t}\]

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it looks tough

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