Question

The solution of the differential equation \(\frac{dy}{dx} = \frac{a+bx}{c+dy}\) represents a family of circles centered at the origin if...

• A. \(a = c = 0, b+d = 0\)
• B. \(a = c = 0, b = d\)
• C. \(b = d = 0, a+c = 0\)
• D. \(b = d = 0, a = c\)

Hint:

Separate variables, integrate, and demand no linear terms and equal x^2, y^2 coefficients.

Answer 1

The Correct Option is A

Step 1: Separate the Variables:
\((c+dy)\,dy=(a+bx)\,dx\).

Step 2: Integrate:
\[ cy+\frac d2y^2=ax+\frac b2x^2+k \Rightarrow \frac b2x^2-\frac d2y^2+ax-cy+k=0 \]

Step 3: Conditions for a Circle at the Origin:
For a circle with centre at the origin there must be no linear terms: \(a=0\) and \(c=0\). Also the coefficients of \(x^2\) and \(y^2\) must be equal: \(\dfrac b2=-\dfrac d2\), i.e. \(b+d=0\).

Step 4: Check the Options:
Option (A) \(a=c=0,\ b+d=0\) satisfies both. Option (B) has \(b=d\), which would give a hyperbola. Options (C) and (D) put \(b=d=0\), which removes the quadratic terms and leaves a straight line.

Final Answer:
The condition is \(a=c=0,\ b+d=0\), option (A). \[ \boxed{\text{(A) } a=c=0,\ b+d=0} \]