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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
Solution
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 .
\[\text{ Therefore, the coffecients of }x^2\text{ and }y^2\text{ must be equal . }\]
\[ i . e . - a = b\]
\[ \Rightarrow a = - b\]
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