Question

If $y(x)$ is the solution of the differential equation $x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2,$ and the slope of the curve $y=y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is ____.

Answer 1

Correct Answer: 8

The answer is 8.

Answer 2

Given,
$x d y-\left(y^2-4 y\right) d x=0 \text { for } x>0, y(1)=2,$
\(\frac{dy}{y^2-4y}=\frac{dx}{x}\)

\(\frac{1}{4}(\frac{1}{y-4}-\frac{1}{y})dy=\frac{dx}{x}\)

Integrate both sides
\(\log_c|y-4|-\log_c|y|=4\log_ex+\log_ec\)
\(|\frac{y-4}{y}|=x^4\)
\(|y-4|=|y|x^4\)
\(y-4=yx^4\)
\(y-yx^4=4\)
\(y=\frac{4}{1-x^4}\)
y(1)=2

\(y(\sqrt2)=\frac{4}{5}\)
\(10y(\sqrt2)=8\)

So, the answer is 8.