Question

The solution of the differential equation\(\frac{dy}{dx}= \frac{6}{x^2}; y(1) = 3\) is:

• A. \(y=9+\frac{6}{x}\)
• B. \(y=9-\frac{6}{x}\)
• C. \(y=-9+\frac{6}{x}\)
• D. \(y=-3-\frac{6}{x}\)

Answer 1

The Correct Option is B

To solve the differential equation \(\frac{dy}{dx}=\frac{6}{x^2}\) with the initial condition \(y(1)=3\), proceed as follows:

First, separate the variables to integrate:

\[\int dy = \int \frac{6}{x^2} dx\]

Integrating both sides, we get:

\[y = -\frac{6}{x} + C\]

Next, use the initial condition \(y(1) = 3\) to find the constant \(C\):

\[3 = -\frac{6}{1} + C\]

Simplifying, we have:

\[3 = -6 + C\]

Thus, \(C = 9\).

The particular solution is:

\[y = 9 - \frac{6}{x}\]

The correct answer is \(y=9-\frac{6}{x}\).