Q29E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Prove that if ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ are linearly independent solutions of  $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{py}\mathbf{\text{'}}\mathbf{+}\mathbf{qy}\mathbf{=}\mathbf{0}$ on $(\mathbf{a}\mathbf{,}\mathbf{b})$, then they cannot both be zero at the same point  ${\mathbf{t}}_{\mathbf{0}}$ in $\left(a,b\right)$

Step-by-Step Solution

Answer

${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ cannot be zero at the same point ${\mathrm{t}}_0$ in $\left(\mathrm{a},\mathrm{b}\right)$.

Step 1: Check linear independence.

Let  ${\mathrm{y}}_1$ and ${\mathrm{y}}_2$ are two independent solutions of the given differential equations, then ${\mathrm{c}}_1{\mathrm{y}}_1+{\mathrm{c}}_2{\mathrm{y}}_2=0$ only if  ${\mathrm{c}}_1={\mathrm{c}}_2=0$.

Step 2: Check whether y 1 , y 2 can be zero or not.

If  ${\mathrm{y}}_1\left({\mathrm{t}}_0\right)={\mathrm{y}}_2\left({\mathrm{t}}_0\right)=0$, then ${\mathrm{c}}_1{\mathrm{y}}_1\left({\mathrm{t}}_0\right)+{\mathrm{c}}_2\mathrm{y}\left({\mathrm{t}}_0\right)=0$ even if  ${\mathrm{c}}_1={\mathrm{c}}_2\ne 0$.

 

Thus, it contradicts the linear independence of the solutions. Therefore ${\mathbf{y}}_{\mathbf{1}}$ and ${\mathbf{y}}_{\mathbf{2}}$ cannot be zero at the same point.