Pls-i-need-a-help-from-Ordinary-differantial-Equation-t-2-y-2-t-t-y-1-t-t-2-2-y-t-0-we-Already-Know-Solution-y-t-C-1-J-t-C-2-J-t-But-J-t-can-t-Satisfy-as-Solutio

Question Number 212906 by issac last updated on 26/Oct/24

Pls i need a help . .

from Ordinary differantial Equation

t^{2} y^{(2)} (t) + t \cdot y^{(1)} (t) + (t^{2} - ν^{2}) y (t) = 0

we Already Know

Solution y (t) = C_{1} J_{ν} (t) + C_{2} J_{- ν} (t)

But J_{- ν} (t) {can}^{'} t Satisfy as Solution

Cus J_{ν} (t) and J_{- ν} (t) are Not Linear independent .

Wronskian W \in mat (m, m)

\det W = 0

thus Solution y (t) = C_{1} J_{ν} (t) + C_{2} Y_{ν} (t)

i already undertand above indentity

i wrote

my question is prove {Abel}^{'} s identity

W (J_{ν} (t), Y_{ν} (t)) = \frac{2}{π t}

Pls Help : (

Commented by issac last updated on 26/Oct/24

Wronskian mat (n, n)

W = (\begin{matrix} f_{1} & f_{2} & \dots & f_{n} \\ f_{1}^{(1)} & f_{2}^{(1)} & \dots & f_{n}^{(1)} \\ ⋮ & ⋮ \\ f_{1}^{(m - 1)} & f_{2}^{(m - 1)} & \dots & f_{n}^{(m - 1)} \end{matrix})

\det W = 0 \to linear dependence .

\det W \neq 0 \to linear independence .

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