The particular solution of the differential equation \(\displaystyle\frac{dy}{dx}+\frac{3y}{x}= 0, y(1) = 1\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(y=\displaystyle\frac{1}{x^3}\)
The equation is in variables separable form:
\( \displaystyle \frac{dy}{dx} = -\frac{3y}{x} \)
\( \displaystyle \frac{dy}{y} = -3\,\frac{dx}{x} \)
Integrating both sides:
\(\log|y| = -3\log|x| + \log|C|\)
\(\log|y| = \log\left|C x^{-3}\right|\)
\( \displaystyle y = \frac{C}{x^3} \)
Using the initial condition \(y(1) = 1\):
\( \displaystyle 1 = \frac{C}{1},\quad C = 1 \)
\( \displaystyle y = \frac{1}{x^3} \)