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

Answer: (C)

-2

Answer 2

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 $a = −2.$