Question

Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]

Function y = log x

Answer 1

We have,

\[y = \log x...........(1)\]

Differentiating both sides of (1) with respect to x, we get

\[\frac{dy}{dx} = \frac{1}{x}\]

or,

\[x\frac{dy}{dx} = 1\]

It is the given differential equation.

Thus, \[y = \log x\]  satisfies the given differential equation.

Hence, it is a solution.

Also, when \[x = 1, y = \log 1 = 0, i.e., y\left( 1 \right) = 0\]

Hence, \[y = \log x\] is the solution to the given initial value problem.