Question

pls explain the concept of homogeneous differential equation

Answer 1

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

Answer 2

how do u solve it

Answer 3

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:
\[y(t)=c_1e^{-2t}+c_2e^{-t}\]

Answer 4

it looks tough