Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
The solution of $(y(1 + x^{-1}) + \sin y)dx + (x + \log_e x + x\cos y)dy = 0$ is:
(1 + y^{-1}\sin y) + x^{-1}\log_e x = C$
$(y + \sin y) + xy\log_e x = C$
$xy + y\log_e x + x\sin y = C$
None of these
Step-by-Step Solution
Key Concept: Grouping terms strategically reveals exact differential combinations $d(uv)$ that can be integrated directly.
Rewrite the given equation as $y(1+x^{-1})dx + (x+\log x)dx + \sin y dx + x\cos y dy = 0$. Group terms to obtain $d(y(x+\log x)) + d(x\sin y) = 0$, which integrates to $y(x+\log x) + x\sin y = C$.
Correct Answer: 3