Can anyone explain why is there complex solutions to linear inhomogeneous differential equations? I understand that there is a homogeneous solution + a particular inhomogeneous soultion and the sum of those are the solution to inhomogeneous differential equations, but i dont feel it :D , if we solve an equation, the steps that we make, lead us to the solution, and for example if we solve an inhomogeneous differential equation with method of undetermined coefficients, we will get a solution, what would seem fullfill the equation, but there will be also a homogeneous solution, how does it comes,between the ordinary equations, we cant make such a plus solution, sorry but i dont feel intuitively this
 A: Understanding why we need to add the homogeneous solution to the particular solution to get the full solution to a differential equation may seem a bit confusing.  This is very reasonable since we are most comfortable studying algebraic equations like $ax = b$.  You would agree that (if $b \ne 0$), $a(0+\frac{b}{a}) = b$ so that the solution is $x = 0 + \frac{b}{a} = \frac{b}{a}$.  So, for very basic linear algebraic equations, we can ignore the $0$, since it does not add anything to the solution.  However, in differential equations, the homogeneous solutions acts like a $0$, except that it does add to the solution.  The homogeneous solution is like the additive identity for differential operators.
A: If you have a typical inhomogeneous equation like $$y''+k^2y=f(x)$$ and find a solution $Y(x)$ you can just plug in $Y(x)+a\sin (kx)+b\cos(kx)$ and see it is a solution as well.  That is because $\sin(kx)$ and $\cos (kx)$ are both solutions of the homogeneous part.  You can think of it as the left side not being sensitive to these functions, so you can add any multiple to one solution and get another solution.  
For homogeneous linear differential equations, you typically have a vector space of solutions with dimension equal to the highest derivative in the equation.
