CBSE Class 12 · 2017 · Differential EquationsLong Answer

Solve the differential equation

x \frac{d y}{d x} + y = x cos x + sin x

, given that

y = 1

when

x = \frac{π}{2}

Answer: $y = sin x$

Solution

Divide by

x

to put it in linear form

\frac{d y}{d x} + \frac{1}{x} y = cos x + \frac{sin x}{x}

.
Integrating factor

= e^{\int (1/ x) d x} = e^{l o g x} = x

. Multiplying through, the left side becomes

\frac{d}{d x} (x y)

\frac{d}{d x} (x y) = x cos x + sin x \Rightarrow x y = \int (x cos x + sin x) d x

Since

\int x cos x d x = x sin x + cos x

and

\int sin x d x = - cos x

, the

cos x

terms cancel:

x y = x sin x + C \Rightarrow y = sin x + \frac{C}{x}

Using

y = 1

x = \frac{π}{2}

1 = 1 + \frac{2 C}{π} \Rightarrow C = 0

. Hence

y = sin x

. Off-chapter: this is a linear first-order differential equation, not a definite integral; filed under Differential Equations.

CBSE Class 12 Board · 2017 · Subjective