Question

The integrating factor of the differential equation xy'+2y-7x³=0 is

• A. log|x|
• B. x²
• C. \(\frac{1}{x^2}\)
• D. \(\frac{1}{2}log|x|\)
• E. x

Answer 1

The Correct Option is B

The given differential equation is \( x y' + 2y - 7x^3 = 0 \). 

We can rewrite this equation as:

\( x y' + 2y = 7x^3 \),

which is a linear first-order differential equation in the form \( y' + P(x) y = Q(x) \), where:

\( P(x) = \frac{2}{x} \) and \( Q(x) = 7x^2 \).

The integrating factor \( \mu(x) \) is given by:

\( \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{2}{x} \, dx} = e^{2 \log |x|} = |x|^2 \).

Thus, the integrating factor is \( x^2 \) (considering \( x > 0 \)).

The correct answer is \( x^2 \).