If I have the differential operator $$L = \dfrac{d}{dx} + c v(x)$$ what will be equal to $L^2$?
To $L^2 v(x)?$
What is the meaning of $L^2$? What are my issues to compute it?
In my task I have the formula $$L^N v(x) + \sum\limits_{j=1}^N a_j(x)(L^{j - 1} v(x)) + a_0(x) = 0,$$ where $N$ is the order of equation. So I need to find $L^N v(x)?$ somehow, but I don't know if I correctly work with the differential operator.
What have I tried?
I tried to get the result by substituting $N=2$: $$L^2 v(x) + \sum\limits_{j=1}^2 a_j(x)(L^{j - 1} v(x)) + a_0(x) = 0,$$ $$(\dfrac{d}{dx} + c v(x))^2 v(x) + a_1(x) v(x) + a_2(x)((\dfrac{d}{dx} + c v(x)) v(x)) + a_0(x) = 0,$$ $$\dfrac{d^2v}{dx^2} + c^2 v^3(x) + 2c v(x) \dfrac{dv}{dx} + a_1(x) v(x) + a_2(x)\dfrac{d}{dx} + a_2(x)c v^2(x) + a_0(x) = 0,$$ $$\dfrac{d^2v}{dx^2} + (a_2(x) + 2c v(x)) \dfrac{dv}{dx} + c^2 v^3(x) + a_1(x) v(x) + a_2(x)c v^2(x) + a_0(x) = 0,$$
But the actual result is: $$\dfrac{d^2v}{dx^2} + (a_2(x) + \color{red}{3}c v(x)) \dfrac{dv}{dx} + c^2 v^3(x) + a_1(x) v(x) + a_2(x)c v^2(x) + a_0(x) = 0,$$ where somehow I get $\color{red}{3}c v(x)$. So I guess I work incorctly with differential operator $L$.
In general the operator $L^2$ is $$L^2=\left(\dfrac{\mathrm{d}}{\mathrm{d}x}+cv(x)\right)^2=\dfrac{\mathrm{d}^2}{\mathrm{d}x^2}+cv(x)\dfrac{\mathrm{d}}{\mathrm{d}x}+c\dfrac{\mathrm{d}}{\mathrm{d}x}v(x)+c^2v(x)^2$$ So when you apply it to $v(x)$ you have $$\left(\dfrac{\mathrm{d}^2}{\mathrm{d}x^2}+cv(x)\dfrac{\mathrm{d}}{\mathrm{d}x}+c\dfrac{\mathrm{d}}{\mathrm{d}x}v(x)+c^2v(x)^2\right)v(x)$$ $$v''(x)+2cv(x)v'(x)+c^2v(x)^3$$