If $x=a\cos\theta+b\sin\theta,\ y=a\sin\theta-b\cos\theta$, show that $y^{2}\dfrac{d^{2}y}{dx^{2}}-x\dfrac{dy}{dx}+y=0$.

CBSE Class 12 · 2015 · DifferentiationLong Answer

x = a cos θ + b sin θ, y = a sin θ - b cos θ

, show that

y^{2} \frac{d ^{2} y}{d x ^{2}} - x \frac{d y}{d x} + y = 0

Answer: Proved

Square and add to eliminate

θ

x^{2} + y^{2} = a^{2} + b^{2} (constant)

Differentiate w.r.t.

x

2 x + 2 y \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x}{y}

Differentiate again (quotient rule), then substitute

\frac{d y}{d x} = - x / y

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

Now form the required expression:

y^{2} \frac{d ^{2} y}{d x ^{2}} - x \frac{d y}{d x} + y = - \frac{x ^{2} + y ^{2}}{y} + \frac{x ^{2}}{y} + y = \frac{- x ^{2} - y ^{2} + x ^{2} + y ^{2}}{y} = 0

y^{2} \frac{d ^{2} y}{d x ^{2}} - x \frac{d y}{d x} + y = 0

CBSE Class 12 Board · 2015 · Subjective