Chapter 6.2, Problem 17P

Verify that the differential operator defined by

$L\left[y\right]=y^{\left(n\right)}+p_1\left(t\right)y^{\left(n-1\right)}+\mathrm{....}+p_n\left(t\right)y$

is a linear operator. That is show that

$L\left[c_1{y}_1+c_2{y}_2\right]=c_{1}L\left[y_1\right]+c_{2}L\left[y_2\right]$.

where $y_1$ and $y_2$ are $n$ times differentiable functions and $c_1$ and $c_2$ are arbitrary constants. Hence show that if $y_1,y_2,\mathrm{.....}{y}_n$ are solutions of $L\left[y\right]=0$, then the linear combination $c_1{y}_1+\mathrm{......}+c_n{y}_n$ is also solution of $L\left[y\right]=0$.