Question

Solution of \( (2y - x) \frac{dy}{dx} = 1 \) is

• A. \( x = 2(y - 1) + ce^{-y} \), where c is the constant of integration
• B. \( x = 2(y - 1) + ce^{-x} \), where c is the constant of integration
• C. \( y = 2(x - 1) + ce^{-x} \), where c is the constant of integration
• D. \( y = 2(x - 1) + ce^{-y} \), where c is the constant of integration

Hint:

If $dy/dx$ looks complicated, try inverted form $dx/dy$; it often turns into a standard linear DE.

Answer 1

The Correct Option is A

Step 1: Rewrite as Linear DE
$\frac{dx}{dy} = 2y - x \implies \frac{dx}{dy} + x = 2y$. This is a linear differential equation in $x$.
Step 2: Find Integrating Factor (I.F.)
$P = 1 \implies I.F. = e^{\int 1 dy} = e^y$.
Step 3: Solve
$x \cdot e^y = \int 2y e^y dy$. Using Integration by Parts: $\int y e^y dy = y e^y - e^y$. $x e^y = 2(y e^y - e^y) + c$.
Step 4: Simplify
$x = 2y - 2 + c e^{-y} = 2(y-1) + c e^{-y}$.
Final Answer:(A)