If x-t-1 and y-t3-3t- then d2ydx2 is equal to

We have, x=t+1 ⇒dxdt=1 and, y=t^3+3t ⇒dydt=3t^2+3 ∴dydx=dydtdxdt ⇒dydx=3t^2+31=3(t^2+1) ⇒d^2ydx^2=ddx(dydx)=ddx(3(t^2+1)) ⇒d^2ydx^2=3ddt(t^2+1)×dtdx ⇒d^2ydx^2=6 ...

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Byju's Answer

Standard XII

Mathematics

Order of a Differential Equation

If x=t+1 and ...

Question

If $x = t + 1$ and $y = t^{3} + 3 t,$ then $\frac{d^{2} y}{d x^{2}}$ is equal to

t^{2}

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6 t

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t^{3}

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6 t^{3}

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Solution

The correct option is B $6 t$
We have,
$x = t + 1$
$\Rightarrow \frac{d x}{d t} = 1$
and, $y = t^{3} + 3 t$
$\Rightarrow \frac{d y}{d t} = 3 t^{2} + 3$
$∴ \frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
$\Rightarrow \frac{d y}{d x} = \frac{3 t^{2} + 3}{1} = 3 (t^{2} + 1)$
$\Rightarrow \frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (3 (t^{2} + 1))$
$\Rightarrow \frac{d^{2} y}{d x^{2}} = 3 \frac{d}{d t} (t^{2} + 1) \times \frac{d t}{d x}$
$\Rightarrow \frac{d^{2} y}{d x^{2}} = 6 t \times 1 = 6 t$

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If x=t+1 and y=t3+3t, then d2ydx2 is equal to

The correct option is B 6tWe have, x=t+1 ⇒dxdt=1 and, y=t3+3t ⇒dydt=3t2+3 ∴dydx=dydtdxdt ⇒dydx=3t2+31=3(t2+1) ⇒d2ydx2=ddx(dydx)=ddx(3(t2+1)) ⇒d2ydx2=3ddt(t2+1)×dtdx ⇒d2ydx2=6t×1=6t

If $x = t + 1$ and $y = t^{3} + 3 t,$ then $\frac{d^{2} y}{d x^{2}}$ is equal to