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Question Number 16475 by Sai dadon. last updated on 22/Jun/17

u s e L {f^{'} (t)} t o f i n d l a p l a c e

t r a n s f o r m o f d^{2} x / {d t}^{2} + n^{2} x = k c o s w t

g i v e n t h a t x = 0 a n d d x / d t = 0 w h e n t = 0

P L Z H E L P .

Answered by sma3l2996 last updated on 23/Jun/17

w e k n o w t h a t L {f^{'} (t)} = s L {f (t)} - f (0) a n d L {f^{″} (t)} = s L {f^{'} (t)} - f^{'} (0)

s o L {f^{″} (t)} = s^{2} L {f (t)} - s f (0) - f^{'} (0)

l e t x (t) = f (t)

L {\frac{d^{2} x (t)}{{d t}^{2}}} = s^{2} L {x (t)} - s x (0) - \frac{d x (0)}{d t} = s^{2} L {x (t)}

a n d a l s o w e h a v e :

L {\frac{d^{2} x}{{d t}^{2}} + n^{2} x} = L {k c o s (w t)}

\Leftrightarrow L {\frac{d^{2} x}{{d t}^{2}}} + n^{2} L {x} = k L {c o s (w t)} = k . \frac{s}{s^{2} + w^{2}}

s^{2} L {x} + n^{2} L {x} = k . \frac{s}{s^{2} + w^{2}}

(s^{2} + n^{2}) . L {x} = k . \frac{s}{s^{2} + w^{2}}

L {x} = k . \frac{s}{(s^{2} + w^{2}) (s^{2} + n^{2})}

\frac{s}{(s^{2} + w^{2}) (s^{2} + n^{2})} = \frac{a s + b}{s^{2} + w^{2}} + \frac{c s + d}{s^{2} + n^{2}}

a = - c, \frac{b}{w^{2}} = - \frac{d}{n^{2}} \Leftrightarrow d = - {(\frac{n}{w})}^{2} b

\frac{1}{(1 + w^{2}) (1 + n^{2})} = \frac{a + b}{1 + w^{2}} - \frac{a + {(\frac{n}{w})}^{2} b}{1 + n^{2}} \Leftrightarrow (a + b) (1 + n^{2}) - (a + {(\frac{n}{w})}^{2} b) (1 + w^{2}) = 1

a (n^{2} - w^{2}) + b (1 + n^{2} - {(\frac{n}{w})}^{2} (1 + w^{2})) = 1

a (n^{2} - w^{2}) = 1 + b (\frac{n^{2} - w^{2}}{w^{2}})

(i) : a = \frac{b}{w^{2}} + \frac{1}{n^{2} - w^{2}}

\frac{2}{(w^{2} + 4) (n^{2} + 4)} = \frac{2 a + b}{w^{2} + 4} - \frac{2 a + {(\frac{n}{w})}^{2} b}{n^{2} + 4}

(2 a + b) (n^{2} + 4) - (2 a + {(\frac{n}{w})}^{2} b) (w^{2} + 4) = 2

2 a (n^{2} - w^{2}) + b (4 - 4 {(\frac{n}{w})}^{2}) = 2

4 b (\frac{w^{2} - n^{2}}{w^{2}}) = 2 + 2 a (w^{2} - n^{2})

(i i) : \frac{b}{w^{2}} = \frac{1}{2 (w^{2} - n^{2})} + \frac{a}{2}

f r o m (i) a n d (i i)

a = \frac{1}{2} a + \frac{1}{2 (w^{2} - n^{2})} + \frac{1}{n^{2} - w^{2}}

a = \frac{1}{n^{2} - w^{2}} = - c

\frac{b}{w^{2}} = \frac{1}{2 (w^{2} - n^{2})} + \frac{1}{2 (n^{2} - w^{2})} = 0

a = - c = \frac{1}{n^{2} - w^{2}} a n d b = d = 0

s o : L {x} = \frac{k}{n^{2} - w^{2}} (\frac{s}{s^{2} + w^{2}} - \frac{s}{s^{2} + n^{2}}) = \frac{k}{n^{2} - w^{2}} (L {c o s (w t)} - L {c o s (n t)})

L {x (t)} = L {\frac{k}{n^{2} - w^{2}} (c o s (w t) - c o s (n t))}

s o : x (t) = \frac{k (c o s (w t) - c o s (n t))}{n^{2} - w^{2}}