The solution of the initial value problem \( x\displaystyle\frac{dy}{dx}=2 \), \( y(1)=2 \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( y=2(\log|x|+1) \)
The variables separate:
\( \displaystyle dy=\frac{2}{x}\,dx \)
\( \displaystyle y=2\log|x|+C \)
Now use \( y(1)=2 \). Since \( \log 1=0 \):
\( \displaystyle 2=2(0)+C\ \Rightarrow\ C=2 \)
\( \displaystyle y=2\log|x|+2 \)
\( \displaystyle =2(\log|x|+1) \)
Shortcut: Check both conditions on the options: y must be 2 at x = 1, and \( x\frac{dy}{dx} \) must equal 2. Only Option (1) passes both.