If $(a+bx)e^{y/x}=x$, then prove that $x^{3}\dfrac{d^{2}y}{dx^{2}}=\left(x\dfrac{dy}{dx}-y\right)^{2}$.

CBSE Class 12 · 2019 · DifferentiationLong Answer

(a + b x) e^{y / x} = x

, then prove that

x^{3} \frac{d ^{2} y}{d x ^{2}} = (x \frac{d y}{d x} - y)^{2}

Answer: Proved

Take logarithm:

lo g (a + b x) + \frac{y}{x} = lo g x

, i.e.

\frac{y}{x} = lo g x - lo g (a + b x)

.
Differentiate w.r.t.

x

(note

\frac{d}{d x} (y / x) = \frac{x y ^{'} - y}{x ^{2}}

\frac{x \frac{d y}{d x} - y}{x ^{2}} = \frac{1}{x} - \frac{b}{a + b x} = \frac{a}{x ( a + b x )}

x \frac{d y}{d x} - y = \frac{a x}{a + b x}

. Differentiate this relation once more; the left side gives

x \frac{d ^{2} y}{d x ^{2}}

x \frac{d ^{2} y}{d x ^{2}} = \frac{a ( a + b x ) - a x \cdot b}{( a + b x ) ^{2}} = \frac{a ^{2}}{( a + b x ) ^{2}}

Therefore

x^{3} \frac{d ^{2} y}{d x ^{2}} = \frac{a ^{2} x ^{2}}{( a + b x ) ^{2}} = (\frac{a x}{a + b x})^{2} = (x \frac{d y}{d x} - y)^{2}

x^{3} \frac{d ^{2} y}{d x ^{2}} = (x \frac{d y}{d x} - y)^{2}

CBSE Class 12 Board · 2019 · Subjective