Question

If the solution of the differential equation \[ \frac{dy}{dx} = \frac{ax + 3}{2y + 5} \] represents a circle, then $a$ is equal to:

• A. 3
• B. -3
• C. -2
• D. 5

Answer 1

The Correct Option is C

Solution: To determine \(a\), solve the given differential equation and check the conditions under which the solution represents a circle.

Rewrite the differential equation: The given equation is:

\[\frac{dy}{dx}=\frac{ax+3}{2y+5}.\]

Separating variables:

\[(2y+5)dy=(ax+3)dx.\]

Integrate both sides: Integrating the left-hand side:

\[\int(2y+5)dy=\int2y~dy+\int5~dy=y^{2}+5y+C_{1},\] where \(C_{1}\) is the constant of integration.

Integrating the right-hand side:

\[\int(ax+3)dx=\int ax~dx+\int3~dx=\frac{ax^{2}}{2}+3x+C_{2},\] where \(C_{2}\) is another constant of integration.

Equating the two sides:

\[y^{2}+5y=\frac{ax^{2}}{2}+3x+C,\] where \(C=C_{2}-C_{1}.\)

Rearrange to standard form: To represent a circle, the equation must take the form:

\[(x-h)^{2}+(y-k)^{2}=r^{2}.\]

The \(y^{2}\) term is already present, but for the \(x^{2}\) term to have the same coefficient as \(y^{2}\), \(a\) must satisfy:

\[\frac{a}{2}=1\Rightarrow a=2.\]
Thus, the value of \(a\) that makes the solution represent a circle is \(a = −2.\)