d-2-x-dt-2-a-b-1-l-x-2-l-2-x-Find-x-t-if-x-0-x-0-x-0-0-

Question Number 40920 by ajfour last updated on 29/Jul/18

\frac{d^{2} x}{{d t}^{2}} = a - b (1 - \frac{l}{\sqrt{x^{2} + l^{2}}}) x

F i n d x (t) i f x (0) = x_{0}, x^{'} (0) = 0 .

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Jul/18

v \frac{d v}{d x} = a - b (1 - \frac{l}{\sqrt{x^{2} + l^{2}}}) x

v d v = (a - b x + \frac{b l x}{\sqrt{x^{2} + l^{2}}}) d x

v d v = (a - b x) d x + \frac{b l}{2} \frac{d (x^{2} + l^{2})}{\sqrt{x^{2} + l^{2}}}

i n t r e g a t i n g

\frac{v^{2}}{2} = a x - b \frac{x^{2}}{2} + \frac{b l}{2} \frac{\sqrt{x^{2} + l^{2}}}{\frac{1}{2}} + c

v = \frac{d x}{d t} = x^{'}

\frac{v^{2}}{2} = a x - \frac{{b x}^{2}}{2} + b l \sqrt{x^{2} + l^{2}} + c

b o u n d a r y c o n d i t i o n

a t t = 0 \frac{d x}{d t} = v = 0 x = x_{0}

0 = {a x}_{0} - \frac{{b x}_{0}^{2}}{2} + b l \sqrt{x_{0}^{2} + l^{2}} + c

\frac{v^{2}}{2} = a (x - x_{0}) - \frac{b^{2}}{2} (x^{2} - x_{0}^{2}) + b l (\sqrt{x^{2} + l^{2}} - \sqrt{x_{0}^{2} + l^{2}}

c o n t d \dots

Commented by ajfour last updated on 29/Jul/18

t h a n k s s i r f o r a s f a r . .

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Jul/18

\sqrt{2 {(\frac{- b}{2} x^{2} + a x + \frac{b^{2}}{2} x_{0}^{2} - {a x}_{0} - b l \sqrt{x_{0}^{2} + l^{2}} + b l \sqrt{x^{2} + l^{2}}}

t h e w h o l e e x p r e s s i o n = \frac{d x}{d t}

\frac{d x}{d t} = \sqrt{2 ({A x}^{2} + B x + C + D \sqrt{x^{2} + l^{2}}} f o r m

x = \int \sqrt{2 ({A x}^{2} + B x + C + D \sqrt{x^{2} + l^{2}}} d t