find-all-values-of-m-R-such-that-the-equation-0-x-arctany-y-dy-mx-has-two-real-roots-x-1-0-x-2-0-

Question Number 180856 by Shrinava last updated on 18/Nov/22

find all values of m \in R such that the equation :

\int_{0}^{x} \frac{\arctan y}{y} dy = mx

has two real roots : x_{1} \in (- \infty; 0), x_{2} \in (0; \infty)

Answered by mr W last updated on 19/Nov/22

f (x) = \int_{0}^{x} \frac{\tan^{- 1} t}{t} d t

g (x) = m x

w e c a n s e e f (x) i s a n o d d f u n c t i o n,

i . e . f (- x) = - f (x) .

f^{'} (x) = \frac{\tan^{- 1} x}{x}

f^{'} (0) = \lim_{x \to 0} \frac{\tan^{- 1} x}{x} = 1

i . e . t h e t a n g e n t o f y = f (x) a t x = 0 i s

y = k x w i t h k = 1 .

\lim_{x \to + \infty} f^{'} (x) = \lim_{x \to + \infty} \frac{\tan^{- 1} x}{x} = 0

s o w e c a n k n o w h o w t h e g r a p h o f

y = f (x) n e a r l y l o o k s l i k e . t h e l i n e

y = m x a l w a y s i n t e r s e c t s t h e c u r v e

y = f (x) a t x = 0 a n d a d d i t i o n a l l y a t

t w o p o i n t s P a n d P^{'} w i t h x = \pm a

i f y = m x i s f l a t e r t h a n t h e t a n g e n t

l i n e a t x = 0, i . e . i f 0 < m < k = 1 .

t h a t m e a n s i f 0 < m < 1, t h e e q n .

f (x) = g (x) w i l l h a v e t w o r o o t s :

x = \pm a . s o t h e a n s w e r i s 0 < m < 1 .

Commented by mr W last updated on 19/Nov/22

Commented by mr W last updated on 19/Nov/22

f o r t h e s o l i t i o n o f t h e q u e s t i o n, w e

{d o n}^{'} t n e e d t o s o l v e t h e i n t e g r a l

\int_{0}^{x} \frac{\tan^{- 1} t}{t} d t r e a l l y . t o b e h o n e s t, i {c a n}^{'} t

s o l v e i t . b u t {i t}^{'} s e n o u g h t o k n o w h o w

i t s g r a p h n e a r l y l o o k s l i k e .

Commented by Shrinava last updated on 19/Nov/22

perfect dear professor thank you