If $x\cos(a+y)=\cos y$, then prove that $\dfrac{dy}{dx}=\dfrac{\cos^{2}(a+y)}{\sin a}$ and hence show that $\sin a\dfrac{d^{2}y}{dx^{2}}+\sin 2(a+y)\dfrac{dy}{dx}=0$.

CBSE Class 12 · 2016 · Differential EquationsLong Answer

x cos (a + y) = cos y

, then prove that

\frac{d y}{d x} = \frac{cos ^{2} ( a + y )}{sin a}

and hence show that

sin a \frac{d ^{2} y}{d x ^{2}} + sin 2 (a + y) \frac{d y}{d x} = 0

Answer: Proved

From

x = \frac{cos y}{cos ( a + y )}

, differentiate w.r.t.

y

\frac{d x}{d y} = \frac{cos ( a + y ) ( - sin y ) - cos y ( - sin ( a + y ))}{cos ^{2} ( a + y )} = \frac{sin ( ( a + y ) - y )}{cos ^{2} ( a + y )} = \frac{sin a}{cos ^{2} ( a + y )}

Therefore

\frac{d y}{d x} = \frac{cos ^{2} ( a + y )}{sin a}

Differentiate this w.r.t.

x

(with

sin a

constant):

sin a \frac{d ^{2} y}{d x ^{2}} = 2 cos (a + y) (- sin (a + y)) \frac{d y}{d x} = - sin 2 (a + y) \frac{d y}{d x}

sin a \frac{d ^{2} y}{d x ^{2}} + sin 2 (a + y) \frac{d y}{d x} = 0

CBSE Class 12 Board · 2016 · Subjective