Q-solve-if-t-2-n-2-cos-2-x-m-2-sin-2-x-then-show-that-t-d-2-t-dx-2-nm-2-t-3-

Question Number 79513 by M±th+et£s last updated on 25/Jan/20

Q . s o l v e

i f t^{2} = n^{2} {c o s}^{2} (x) + m^{2} {s i n}^{2} (x)

t h e n s h o w t h a t :

t + \frac{d^{2} t}{{d x}^{2}} = \frac{{(n m)}^{2}}{t^{3}}

Commented by john santu last updated on 26/Jan/20

t^{2} = n^{2} (1 - \sin^{2} x) + m^{2} \sin^{2} x

t^{2} = n^{2} + (m^{2} - n^{2}) \sin^{2} x

2 t \frac{d t}{d x} = (m^{2} - n^{2}) \sin (2 x)

\frac{d t}{d x} = \frac{(m^{2} - n^{2}) \sin (2 x)}{2 t}

2 {(\frac{d t}{d x})}^{2} + 2 t \frac{d^{2} t}{{d x}^{2}} = 2 (m^{2} - n^{2}) \cos (2 x)

\frac{{(m^{2} - n^{2})}^{2} \sin^{2} (2 x)}{4 t^{2}} + t (\frac{d^{2} t}{{d x}^{2}}) = (m^{2} - n^{2}) \cos (2 x)

c o n t i n u e \dots