Question:
\(a,b,c,d\) are real numbers. The general solution of
\[
\frac{dy}{dx}=\frac{ax+b}{cy+d}
\]
represents a family of straight lines, when
\(a,b,c,d\) are real numbers. The general solution of
\[
\frac{dy}{dx}=\frac{ax+b}{cy+d}
\]
represents a family of straight lines, when
Show Hint
A differential equation represents a family of straight lines when its slope becomes constant, that is, when \(\frac{dy}{dx}=m\).
Updated On: Jun 26, 2026
- \(a=c=0,\ b^2+d^2\neq 0\)
- \(a\neq 0,\ c=0\) or \(a=0,\ c\neq 0\)
- \(bd=0,\ a\neq 0,\ c\neq 0\)
- \(b+d=0,\ a+c=0\)
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The Correct Option is A
Solution and Explanation
Step 1: Analyze the given differential equation.
The given equation is \[ \frac{dy}{dx}=\frac{ax+b}{cy+d} \] For the general solution to represent a family of straight lines, the slope must be constant.
Step 2: Make the right-hand side constant.
If \[ a=0 \] and \[ c=0, \] then the equation becomes \[ \frac{dy}{dx}=\frac{b}{d} \] This is a constant slope, provided \(b\) and \(d\) are not both zero.
Hence, \[ b^2+d^2\neq 0 \]
Step 3: Find the general solution.
Now, \[ \frac{dy}{dx}=\frac{b}{d} \] Integrating, \[ y=\frac{b}{d}x+C \] This represents a family of straight lines.
Step 4: Final conclusion.
Therefore, the required condition is \[ \boxed{a=c=0,\ b^2+d^2\neq 0} \]
The given equation is \[ \frac{dy}{dx}=\frac{ax+b}{cy+d} \] For the general solution to represent a family of straight lines, the slope must be constant.
Step 2: Make the right-hand side constant.
If \[ a=0 \] and \[ c=0, \] then the equation becomes \[ \frac{dy}{dx}=\frac{b}{d} \] This is a constant slope, provided \(b\) and \(d\) are not both zero.
Hence, \[ b^2+d^2\neq 0 \]
Step 3: Find the general solution.
Now, \[ \frac{dy}{dx}=\frac{b}{d} \] Integrating, \[ y=\frac{b}{d}x+C \] This represents a family of straight lines.
Step 4: Final conclusion.
Therefore, the required condition is \[ \boxed{a=c=0,\ b^2+d^2\neq 0} \]
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