Solving Euler-equation alike 2nd order DE with disturbing RHS For a homework problem, I have to solve
$$ t^2 \ddot{x} - 3 t \dot{x} + 3x  = t^2 $$
which seems quite similar to the Euler Equation, which I would know how to solve, apart from the disturbing $ t^2 $ on the RHS instead of a constant $0$.
How does one approach such types of DE? Would it make a difference if there would be a constant $ a \neq 0$ on the RHS?
Thanks for any directions!
 A: In general, an equation of the form:
$$\mathcal{L}(x)=f(t)$$
Where $\mathcal{L}$ denotes some differential operator can be solved by finding the form of a general solution "$y_h(t)$"to the homogeneous equation (where the sum of any two solutions is again a solution):
$$\mathcal{L}(y_h)=0$$
And then finding any particular solution $y_p(t)$ that satisfies the inhomogeneous equation (where the sum of any two solutions is not necessarily another solution):
$$\mathcal{L}(y_p)=f(t)$$
Then the full solution to the original problem will be $x(t) = y_h(t)+y_p(t)$.
Whilst finding solutions to inhomogeneous problems can usually be approached in a formulaic way, finding particular solutions are usually more of a case by case basis. We frequently call the function $f$ the "forcing" of the equation, and it is frequently best to attempt to find a particular solution by looking for something that "looks like" the forcing, plug it into the left hand side and see if it works. 
When the forcing is a polynomial, try plugging in a general polynomial of the same degree into the left hand side $(x=at^2+bt+c)$, and then compare coefficients with the right hand side to determine if there exists some real numbers $(a,b,c)$ such that the equation works.
