Problem 20
By using a suitable substitution, bring the equation 2√(x)…
By using a suitable substitution, bring the equation $2\sqrt{x}\,dy=y\,dx$ to a form with separated variables, and then integrate it, given $y(4)=1$.
Explanation of Method
The equation is already separable; integrate directly and then apply the initial condition to determine the constant.
Step 1: Separate variables
\[\frac{dy}{y}=\frac{dx}{2\sqrt{x}},\qquad x>0.\]
Step 2: Integrate
\[\ln|y|=\sqrt{x}+C \quad\Longrightarrow\quad y=Ce^{\sqrt{x}}.\]
Step 3: Apply the initial condition
With $y(4)=1$: $1=Ce^{\sqrt4}=Ce^2$, so $C=e^{-2}$, giving
\[\boxed{\,y=e^{\sqrt{x}-2}.}\]
Result
Final answer:
\[\boxed{\,y=e^{\sqrt{x}-2},\qquad x>0.}\]