# Thread: differental eq from calc

1. ## differental eq from calc

if x=t^(2)+1 and y=t^3 than d^2y/dt^2=?

2. Originally Posted by AnnaBee123
if x=t^(2)+1 and y=t^3 than d^2y/dt^2=?
Do you need $\displaystyle \frac{d^2y}{dt^2}$ or $\displaystyle \frac{d^2y}{dx^2}~?$

3. Originally Posted by AnnaBee123
if x=t^(2)+1 and y=t^3 than d^2y/dt^2=?

well lets get started....

$\displaystyle \frac{dy}{dt}=3t^2$

Now if we take the derivative of the above with respect to t we get

$\displaystyle \frac{d}{dt}\left(\frac{dy}{dt} \right)=\frac{d}{dt}\left(3t^2 \right)$

$\displaystyle \frac{d^2y}{dt^2}=6t$

4. Originally Posted by Plato
Do you need $\displaystyle \frac{d^2y}{dt^2}$ or $\displaystyle \frac{d^2y}{dx^2}~?$
I was thinking the same as there was an equation given for both

$\displaystyle y=f(t)$ and $\displaystyle x=f(t)$

Therefore a solution could be

$\displaystyle \frac{d^2y}{dx^2} = \frac{\tfrac{d^2y}{dt^2}}{\tfrac{d^2x}{dt^2}}$

$\displaystyle = \frac{6t}{2}$

$\displaystyle = 3t$

5. Originally Posted by pickslides
I was thinking the same as there was an equation given for both

$\displaystyle y=f(t)$ and $\displaystyle x=f(t)$

Therefore a solution could be

$\displaystyle \frac{d^2y}{dx^2} = \frac{\tfrac{d^2y}{dt^2}}{\tfrac{d^2x}{dt^2}}$

$\displaystyle = \frac{6t}{2}$

$\displaystyle = 3t$
We need to be careful...

$\displaystyle x=t^2+1$ $\displaystyle y=t^3$

so $\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{3t^2}{2t}=\frac{3}{2}t$

Now $\displaystyle \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\frac{dy}{dx}} {\frac{dx}{dt}}=\frac{\frac{3}{2}}{2t}=\frac{3}{4t }$

Note we can write y as a function of x as follows

$\displaystyle y=(x-1)^{\frac{3}{2}}$

Now if we take two derivatives we get

$\displaystyle \frac{dy}{dx}=\frac{3}{2}(x-1)^{\frac{1}{2}}$

$\displaystyle \frac{d^2y}{dx^2}=\frac{3}{4}\frac{1}{(x-1)^{\frac{1}{2}}}$

Note that since $\displaystyle x=t^2+1$ we get

$\displaystyle \frac{d^2y}{dx^2}=\frac{3}{4}\frac{1}{(x-1)^{\frac{1}{2}}}=\frac{3}{4}\frac{1}{(t^2+1-1)^{\frac{1}{2}}}=\frac{3}{4t}$