The general solution of the differential equation \( 2x\displaystyle\frac{dy}{dx}-y=3x^{2} \) for \( x\gt 0 \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( y=x^{2}+C\sqrt{x} \) : where \( C \) is an arbitrary constant
Dividing by 2x gives a linear differential equation:
\( \displaystyle \frac{dy}{dx}-\frac{1}{2x}\,y=\frac{3x}{2} \)
The integrating factor is
\( \displaystyle e^{-\frac{1}{2}\log x}=x^{-1/2} \)
So the solution is
\( \displaystyle y\,x^{-1/2}=\frac{3}{2}\int x^{1/2}dx \)
\( \displaystyle =x^{3/2}+C \)
\( \displaystyle y=x^{2}+C\sqrt{x} \)
Shortcut: Substitute Option (3) in the equation. Since \( y'=2x+\frac{C}{2\sqrt{x}} \):
\( \displaystyle 2x\left(2x+\frac{C}{2\sqrt{x}}\right)-\left(x^{2}+C\sqrt{x}\right) \)
\( \displaystyle =4x^{2}+C\sqrt{x}-x^{2}-C\sqrt{x}=3x^{2} \)