Let \( y=f(x) \) be the solution curve of the differential equation \( \displaystyle\frac{dy}{dx}+2y^{2}=0 \) if \( f(1)=1 \), then \( f(2) \) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{1}{3} \)
The equation is in variables separable form.
\( \displaystyle \frac{dy}{dx}=-2y^{2} \)
\( \displaystyle \frac{dy}{y^{2}}=-2\,dx \)
Integrating both sides:
\( \displaystyle -\frac{1}{y}=-2x+c \)
\( \displaystyle \frac{1}{y}=2x-c \)
Using f(1) = 1:
\( 1=2-c\ \Rightarrow\ c=1 \)
\( \displaystyle y=\frac{1}{2x-1} \)
\( \displaystyle f(2)=\frac{1}{2(2)-1} \)
\( \displaystyle =\frac{1}{3} \)