Question

If

x= a sin(t) - b cos(t) and
y= a cos(t) + b sin(t)
find d²y/dx²

PLEASE ANSWER FAST!!!!​

Answer 1

We can write

dydx=dydtdxdt

Since, x=a(t+sint)

⟹dxdt=a(1+cost)

And y=a(1−cost)

⟹dydt=a[0−(−sint)]=asint

Therefore,

dydx=asinta(1+cost)=2sint2 cost22cos2 t2

=tan(t2)

Thus our first answer:

dydx=tan(t2)

Now for the second

d2ydx2=ddx(dydx)

Thus,

d2ydx2=sec2 (t2)⋅12⋅dtdx

Since,

dtdx=1dxdt=1a(1+cost)

Therefore,

d2ydx2=12sec2(t2)1a(1+cost)

d2ydx2=sec2 t22a⋅2cos2 t2

Hence the final result,

d2ydx2=sec4 t24a

Hope that helps :)

I hope you are understand my solution