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

$$\mathrm{Pls}\mathrm{i}\mathrm{need}\mathrm{a}\mathrm{help}..$$$$\mathrm{from}\mathrm{Ordinary}\mathrm{differantial}\mathrm{Equation}$$$$t^2{y}^{\left(2\right)}\left(t\right)+t\cdot y^{\left(1\right)}\left(t\right)+(t^2-{\nu }^2)y\left(t\right)=0$$$$\mathrm{we}\mathrm{Already}\mathrm{Know}$$$$\mathrm{Solution}y\left(t\right)=C_1{J}_{\nu }\left(t\right)+C_2{J}_{-\nu }\left(t\right)$$$$\mathrm{But}{J}_{-\nu }\left(t\right){\mathrm{can}}^{\prime }\mathrm{t}\mathrm{Satisfy}\mathrm{as}\mathrm{Solution}$$$$\mathrm{Cus}{J}_{\nu }\left(t\right)\mathrm{and}{J}_{-\nu }\left(t\right)\mathrm{are}\mathrm{Not}\mathrm{Linear}\mathrm{independent}.$$$$\mathrm{Wronskian}\mathcal{W}\in \mathrm{mat}(m,m)$$$$\det \mathcal{W}=0$$$$\mathrm{thus}\mathrm{Solution}y\left(t\right)=C_1{J}_{\nu }\left(t\right)+C_2{Y}_{\nu }\left(t\right)$$$$\mathrm{i}\mathrm{already}\mathrm{undertand}\mathrm{above}\mathrm{indentity}$$$$\mathrm{i}\mathrm{wrote}$$$$\mathrm{my}\mathrm{question}\mathrm{is}\mathrm{prove}{\mathrm{Abel}}^{\prime }\mathrm{s}\mathrm{identity}$$$$\mathcal{W}(J_{\nu }\left(t\right),Y_{\nu }\left(t\right))=\frac{2}{\pi t}$$$$\mathbf{Pls}\mathbf{Help}:($$

Commented by issac last updated on 26/Oct/24

$$$$$$\mathrm{Wronskian}\mathrm{mat}(n,n)$$$$\mathcal{W}=\left(\begin{array}{cccc}{f}_1& f_2& \dots & f_n\\ f_1^{\left(1\right)}& f_2^{\left(1\right)}& \dots & f_n^{\left(1\right)}\\ ⋮& & ⋮& \\ f_1^{(m-1)}& f_2^{(m-1)}& \dots & f_n^{(m-1)}\end{array}\right)$$$$\det \mathcal{W}=0\to \mathrm{linear}\mathrm{dependence}.$$$$\det \mathcal{W}\ne 0\to \mathrm{linear}\mathrm{independence}.$$

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