Question

The solution of x² + y²  \[\frac{dy}{dx}\]= 4, is

Options

• x² + y² = 12x + C

• x² + y² = 3x + C

• x³ + y³ = 3x + C

• x³ + y³ = 12x + C

Answer 1

x³ + y³ = 12x + C

 

We have, 
\[ x^2 + y^2 \frac{dy}{dx} = 4\]
\[ \Rightarrow y^2 \frac{dy}{dx} = 4 - x^2 \]
\[ \Rightarrow y^2 dy = \left( 4 - x^2 \right)dx\]
Integrating both sides, we get
\[\int y^2 dy = \int\left( 4 - x^2 \right)dx\]
\[ \Rightarrow \frac{y^3}{3} = 4x - \frac{x^3}{3} + D\]
\[ \Rightarrow y^3 = 12x - x^3 + 3D\]
\[ \Rightarrow x^3 + y^3 = 12x + C,\text{ where }C = 3D\]