If x=ϕt, y=ψt, then d2ydx2 is

We have x=ϕt, y=ψt. Therefore, dydx=dydtdxdt=ψ'ϕ'or d2ydx2=ddxψ'ϕ'=ddtψ'ϕ'dtdx =ϕ'ψ''-ψ'ϕ''ϕ'21ϕ'=ϕ'ψ''-ψ'ϕ''ϕ'3

If x=ϕt, y=ψt, then d2ydx2 is

We have x=ϕt, y=ψt. Therefore, dydx=dydtdxdt=ψ'ϕ'or d2ydx2=ddxψ'ϕ'=ddtψ'ϕ'dtdx =ϕ'ψ''-ψ'ϕ''ϕ'21ϕ'=ϕ'ψ''-ψ'ϕ''ϕ'3

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$If x = ϕ (t), y = ψ (t), then \frac{d^{2} y}{{dx}^{2}} is$

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$\frac{ϕ' ψ'' - ψ' ϕ''}{{(ϕ')}^{2}}$

$\frac{ϕ' ψ'' - ψ' ϕ''}{{(ϕ')}^{3}}$

$\frac{ϕ''}{ψ''}$

$\frac{ψ''}{ϕ''}$

answer is B.

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Detailed Solution

$\begin{array}{l} We have x = ϕ (t), y = ψ (t) . Therefore, \\ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{ψ'}{ϕ'} \\ or \frac{d^{2} y}{{dx}^{2}} = \frac{d}{dx} (\frac{ψ'}{ϕ'}) = \frac{d}{dt} (\frac{ψ'}{ϕ'}) \frac{dt}{dx} \\ = \frac{ϕ' ψ'' - ψ' ϕ''}{{ϕ'}^{2}} \frac{1}{ϕ'} = \frac{ϕ' ψ'' - ψ' ϕ''}{{ϕ'}^{3}} \end{array}$