JEE Main · 2008 · 27 Apr 2008 · Offline · Differential Equations

The solution of the differential equation

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

satisfying the condition

y (1) = 1

Answer: (D)

Solution

Rewrite the equation as

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

and substitute

y = v x

, for which

\frac{d y}{d x} = v + x \frac{d v}{d x}

. This reduces it to

v + x \frac{d v}{d x} = 1 + v

, that is

x \frac{d v}{d x} = 1

, so

d v = \frac{d x}{x}

and

v = ln x + C

. Hence

y = x ln x + C x

, and the condition

y (1) = 1

forces

C = 1

, giving

y = x ln x + x

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option

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