Question

Suppose${\mathit{y}}_{\mathbf{1}}\mathbf{,}{\mathit{y}}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }\mathbf{,}{\mathit{y}}_{\mathbf{k}}$arelinearly independent solutions on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$of a homogeneous linear nth-order differential equation with constant coefficients. By Theorem 4.1.2 it follows that${\mathit{y}}_{\mathbf{k}\mathbf{+}\mathbf{1}}\mathbf{=}\mathbf{0}$is also a solution of the differential equation. Is the set of solutions${\mathit{y}}_{\mathbf{1}}\mathbf{,}{\mathit{y}}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }\mathbf{,}{\mathit{y}}_{\mathbf{k}}\mathbf{,}{\mathit{y}}_{\mathbf{k}\mathbf{+}\mathbf{1}}$ linearly dependent or linearly independent on$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$? Discuss.

Short answer

The set of solutions $y_1,y_2,y_3,\dots ,y_k,y_{k+1}$is linearly dependent on $(-\infty ,\infty )$.

Step-by-step solution

Superposition principle-Homogeneous equations

• In the next theorem we see that the sum, or superposition, of two or more solutions of a homogeneous linear differential equation is also a solution.
• Let $y_1,y_2,\dots ,y_k$be solutions of the homogeneous n-th-order differential equation on an interval I. Then the linear combination$y=c_1{y}_1\left(x\right)+c_2{y}_2\left(x\right)+\cdots +c_k{y}_k\left(x\right),$ Where, the $c_i,i=1,2,\dots ,k$are arbitrary constants, is also a solution on the interval.

Find the solution is linearly dependent or independent

Let us suppose that $y_1,y_2,\dots ,y_k$are k linearly independent solutions$(-\infty ,\infty )$ of a homogeneous linear$n^{th}-$order differential equation with constant coefficients.

By knowing $y_{k+1}=0$,is also a solution of the differential equation.

$y_{k+1}=0$

It can be rewritten as

$\begin{array}{rcl}{y}_{k+1}& =& 0\\ 0·y_1+0·y_2+0·y_3+0·y_4+\dots +1\cdots y_{k+1}& =& 0\end{array}$

Therefore, the set of solutions$y_1,y_2,y_3,\dots ,y_k,y_{k+1}$ is linearly dependent on$(-\infty ,\infty )$ .