Question

The general solution of the differential equation \(\frac{dy}{dx}+\frac{y}{x} = x^2+5\) is ....

• A. \(\frac{x^4}{4}+\frac{5x^2}{2}-xy = c\)
• B. \(\frac{x^4}{4}-\frac{5x^2}{2}-xy = c\)
• C. \(\frac{x^4}{4}-\frac{5x^2}{2}+xy = c\)
• D. \(\frac{x^4}{4}+\frac{5x^2}{2}+xy = c\)

Hint:

Multiply by the integrating factor x to make the left side a perfect derivative.

Answer 1

The Correct Option is A

Step 1: Approach
Verify the answer by differentiating option (A) implicitly.

Step 2: Differentiate
$\dfrac{x^4}{4}+\dfrac{5x^2}{2}-xy=c$ gives $x^3+5x-y-xy'=0$.

Step 3: Rearrange
$xy'+y=x^3+5x$. Dividing by $x$: $y'+\dfrac yx=x^2+5$. This is the given equation, so (A) is the general solution.

Final Answer:
The integrating factor is x and the solution is x^4/4 + 5x^2/2 - xy = c, option (A). \[ \boxed{\frac{x^4}{4}+\frac{5x^2}{2}-xy=c} \]