Suppose $p(x),q(x),r(x)$ are continuous functions on some interval such that $r(x)\neq 0$. How one can prove that the non-homogeneous differential equation $y''+p(x)y'+q(x)y=r(x)$ has exactly two linearly independent solutions? And in general, how do you prove that the maximum number of linearly independent solutions of a differential equation $y^{(n)}+p_{n-1}(x)y^{(n-1)}+\dots + p_1 y'+p_0 y=r(x)$ with continuous coefficients is $n$? I know that in case of a homogeneous differential equation this is true (the solutions form a vector space), but I've never seen a similar statement for non-homogeneous equations.
[Math] Maximum number of linearly independent solutions of nonhomogeneous ODE
ordinary differential equations
Best Answer
A line is determined by 2 points, a (2D) plane needs 3 points to fix it in higher-dimensional space etc.
Which means in general that an affine space of dimension $n$ is determined only by $n+1$ points $y_0,y_1,...,y_n$ and all other points are affine combinations $$ y=c_0y_0+c_1y_1...+c_ny_n\quad\text{ where }\quad c_0+c_1+...+c_n=1 $$ One can eliminate $c_0$ and the coefficient condition via $$ y=y_0+c_1(y_1-y_0)+...+c_n(y_n-y_0) $$ where you find again the usual form of particular solution plus general homogeneous solution with $n$ free parameters.
The $n+1$ inhomogeneous solutions $y_0,y_1,...,y_n$ are in general position, i.e., actually span the full affine inhomogenous solutions space if and only if the homogenous solutions $(y_1-y_0),...,(y_n-y_0)$ are a basis of the homogeneous solution space (which you stated you already know about).