d-2-y-dx-2-c-2-x-2-y-y-a-x-0-

Question Number 73495 by ajfour last updated on 13/Nov/19

\frac{d^{2} y}{{d x}^{2}} = c^{2} x^{2} y (y = a, x = 0)

Answered by mind is power last updated on 13/Nov/19

t = x^{2}

\frac{d y}{d x} = \frac{d y}{d t} . \frac{d t}{d x} = 2 x \frac{d y}{d t}

\frac{d^{2} y}{{d x}^{2}} = 2 \frac{d y}{d t} + 2 x . \frac{d^{2} y}{{d t}^{2}} . \frac{d t}{d x} = 2 \frac{d y}{d t} + 4 x^{2} \frac{d^{2} y}{{d t}^{2}}

\Leftrightarrow 4 x^{2} \frac{d^{2} y}{{d t}^{2}} + 2 \frac{d y}{d t} = c^{2} x^{2} y

\Leftrightarrow 4 t \frac{d^{2} y}{{d t}^{2}} + 2 \frac{d y}{d t} - c^{2} t y (t) = 0

\Leftrightarrow t^{2} \frac{d^{2} y}{{d t}^{2}} + \frac{1}{2} t \frac{d y}{d t} - \frac{c^{2}}{4} t^{2} y (t) = 0

B e s s e l g e n e r a l i s e E q u a t i o n

i s x^{2} \frac{d^{2} y}{{d x}^{2}} + (2 p + 1) x \frac{d y}{d x} + (a^{2} x^{2 r} + β^{2}) y = 0

s o l u t i o n i s y = x^{- p} [C_{1} J_{q / r} (\frac{{a x}^{r}}{r}) + C_{2} Y_{q / r} (\frac{α}{r} x^{r})]

q = \sqrt{p^{2} - β^{2}}

i n o u r c a s e r = 1

a = (\frac{c i}{2})

p = - \frac{1}{4}

q = \sqrt{\frac{1}{16} - 0} = \frac{1}{4}

y = t^{\frac{1}{4}} [C_{1} J_{\frac{1}{4}} (\frac{c i}{2} t) + C_{2} Y_{\frac{1}{4}} (\frac{c i t}{2})]

J f i r s t b a s s e l f u n c t i o n, Y 2 n d e s p a c e o f b a s s e l e q u a t i o n

y (x) = Y (t) ∣_{t = \sqrt{x}}

Commented by ajfour last updated on 13/Nov/19

T h a n k s^{'} P o w e r f u l {M i n d}^{'}, i^{'} l l

l o o k i n t o^{'} E r w i n {K r e y s z i g}^{'}

(h i g h e r e n g g . m a t h s b o o k) a n d

t r y t o u n d e r s t a n d y o u r {s o l}^{n} . .

Commented by mind is power last updated on 13/Nov/19

y^{'} r e W e l c o m i h a d a p d f o f d i f f e r e n t i a l e q u a t i o n

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