QJEEM08Ma4b76: JEE Main 2008 Mathematics - Class 12 JEEM08Ma4b76 Solution: The solution of the differential equation d y d x = x + y x…

Question

Verified Solution • 2025–26 Session moderate

The solution of the differential equation

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

satisfying the condition

y (1) = 1

is :

Accepted Answer

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

Putting

y =

v x

and

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

we get

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

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

\Rightarrow v = \ln x + c

\Rightarrow y = x \ln x + c x

As

y (1) = 1

∴

c = 1

So solution is

y = x l n x + x

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