Multiple solutions of non homogeneous differential equations I understand that the general solution of a non homogeneous ODE is the sum of the solution of the homogeneous ODE plus a particular solution.
However that suggest that there might be different particular solutions for the same ODE, but I never have seen two different particular solutions for the same ODE. Could you give me an example? 
 A: The general solution of the inhomogeneous linear
constant-coefficient second-order differential equation
$$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=f(x)$$
is found as follows:


*

*First find the complementary function $y_c(x)$, i.e. the general solution of the associated homogeneous equation
$$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=0.$$

*Then find a particular integral $y_p(x)$.

*The general solution is then $y_g(x)=y_c(x)+y_p(x)$.


Any choice of particular integral gives the same general solution. Formulas obtained for the general solution may look different for different choices of particular
integral, but they are in fact always equivalent. As an example, consider 
$$\dfrac{d^2y}{dx^2}+9y=9x+9.$$ Its associated homogenuous equation is $$\dfrac{d^2y}{dx^2}+9y=0,$$ which has the solution $$y_c=C\cos 3x + D\sin 3x,$$
where $C$ and $D$ are arbitrary constants. A particular integral is
$$y_p=x+1,$$
therefore the general solution is
$$y_g=y_c+y_p=C\cos 3x + D\sin 3x+x+1.$$
It would have been equally valid to have chosen $$y_p = x + 1+\sin3x$$ as the particular integral. In that case, the general solution would have been obtained as
$$y_g=C\cos 3x + D\sin 3x+x + 1+\sin3x.$$
This form looks a little bit different but it may be written as $$y = C \cos3x + (D + 1)\sin3x + x + 1;$$
and since $C$ and $D$ are arbitrary constants, this form of the general solution represents exactly the same family of solutions.
The same logic applies to inhomogeneous linear constant-coefficient differential equations of first order, third order, etc. Formulas obtained for the general solution are indeed equivalent as differences in the choice of particular integrals are absorbed into the arbitrary coefficients represented by the general solution.
A: Take $y' = y + 1$. Then $y_1 = \mathrm e^x - 1$ is a particular solution, but also $y_2 = 2\mathrm e^x - 1$. 
