IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

ekbana_bombabu

X= 2cost-cos2t , y =2sint-sin2t

Find d^2y/dx^2 at t= pie/4

Y = cos^-1x, find d2y/dx2 in terms of y alone.

Y = 500e^7x+600e^-7x, show that d2y/dx2= 49 y

Y= (tan^-1x)², then prove that (1+x)²y2+2x(1+x²)y1= 2

rini

phanindraramesh

x=2cost-cos2t

y=2sint-sin2t

dx/dt=-2sint+2sin2t

dy/dt=2cost-2cos2t

dy/dx=dy/dt_dx/dt

=2cost-2cos2t/-2sint+2sin2t

=cost-cos2t/-sint+sin2t

problem is simple up to here but here it turns tough

d²y/dx²

=(-sint+sin2t)(-sint*dt/dx+2sin2t*dt/dx)-(cost-cos2t)(-cost*dt/dx+2cos2t*dt/dx)_

(-sint+sin2t)²

=dt/dx[(-sint+sin2t)(-sint+2sin2t)-(cost-cos2t)(-cost+2cos2t)]_(-sint+sin2t)²

putting t=pie/4=45

we will get the answer as 3

2-3/1+

2

see whethere it is correct or not

bye..................

phanindraramesh

second one os easy

y=cos^-1x

cosy=x

diff w.r.t.x

-siny.dy/dx=1

dy/dx=-1/siny

diff w.r.t.x again

d²y/dx²=cosy.dy/dx_sin²y

=cosy.-1/siny_sin²y

=-cosy/sin³y

note :- _ means whole by

i tink problem was solved

bye.................................

phanindraramesh

third one is also easy

y=500e^7x+600e^-7x

differenciating w.r.t. x

dy/dx=500e^{7x *}7+600e^-7x *(-7)

d²y/dx²= 49 .500e^7x+600e^-7x (-7)(-7)

=49(y)