## anonymous 4 years ago pls explain the concept of homogeneous differential equation

1. anonymous

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: $\sin(t)y'(t)+y(t)\cos(3t)e^{2t}=0$

2. anonymous

how do u solve it

3. anonymous

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: $y(t)=c_1e^{-2t}+c_2e^{-t}$

4. anonymous

it looks tough