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 $x=1,y=\log 1=0,\text{i.e.,}y\left(1\right)=0.$
Hence, $y=\log x$ is the solution to the given initial value problem.