Question

If \(x\frac{dy}{dx}=y(\log y-\log x+1)\) and its solution is \(y=[h(x)]e^{cx}\), then \(h(x)=\ldots\)

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

Hint:

Substitute y = vx to reduce it to separable form.

Answer 1

The Correct Option is B

Step 1: Verify the Proposed Solution
Take $y=xe^{cx}$. Then $\dfrac{dy}{dx}=e^{cx}+cxe^{cx}$.

Step 2: Substitute
$x\dfrac{dy}{dx}=xe^{cx}(1+cx)$. Also $\log y-\log x+1=cx+1$, so $y(\log y-\log x+1)=xe^{cx}(1+cx)$.

Step 3: Conclude
Both sides match, so $h(x)=x$.

Final Answer:
The equation is satisfied by y = x e^{cx}, so h(x) = x. \[ \boxed{\text{(B) }x} \]