Question

The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when

Options

• a = b

• a = −b

• a = −2b

• a = 2b

Answer 1

a = −b

 

We have,
\[\frac{dy}{dx} = \frac{ax + g}{by + f}\]
\[ \Rightarrow \left( by + f \right)dy = \left( ax + g \right)dx\]
Integrating both sides, we get
\[\int\left( by + f \right)dy = \int\left( ax + g \right)dx\]
\[ \Rightarrow b\frac{y^2}{2} + fy = a\frac{x^2}{2} + gx + C\]
\[ \Rightarrow b\frac{y^2}{2} + fy - a\frac{x^2}{2} - gx = C\]
\[ \Rightarrow b y^2 + 2fy - a x^2 - 2gx - 2C = 0\]
The above equation represents a circle . 
\[\text{ Therefore, the coffecients of }x^2\text{ and }y^2\text{ must be equal . }\]
\[ i . e . - a = b\]
\[ \Rightarrow a = - b\]