Question

Solve the differential equation \( y \frac{dy}{dx} + x = 0 \)

Hint:

Separate variables for equations like \( y \frac{dy}{dx} = -x \); solution often yields conic sections.

Answer 1

Rewrite: \( y \frac{dy}{dx} = -x \). 
\[ y \, dy = -x \, dx. \] Integrate both sides: 
\[ \int y \, dy = \int -x \, dx. \] \[ \frac{y^2}{2} = -\frac{x^2}{2} + c \Rightarrow y^2 = -x^2 + 2c \Rightarrow x^2 + y^2 = 2c. \] Let \( 2c = k \), where \( k \) is a positive constant: 
\[ x^2 + y^2 = k. \] Answer: \( x^2 + y^2 = k \).