Question:
The solution of the differential equation \(3y\dfrac{dy}{dx}+2x=0\) represents a family of
The solution of the differential equation \(3y\dfrac{dy}{dx}+2x=0\) represents a family of
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If an equation has \(x^2\) and \(y^2\) with same sign and unequal coefficients, it generally represents an ellipse.
Updated On: Jul 3, 2026
- Ellipse
- Circles
- Parabolas
- Hyperbolas
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The Correct Option is A
Solution and Explanation
Concept:
To identify the family of curves represented by a differential equation, solve the differential equation and compare the obtained equation with standard forms of conics. The given equation is \[ 3y\frac{dy}{dx}+2x=0 \]
Step 1: Separate the variables. \[ 3y\frac{dy}{dx}=-2x \] Multiply both sides by \(dx\): \[ 3y\,dy=-2x\,dx \]
Step 2: Integrate both sides. \[ \int 3y\,dy=\int -2x\,dx \] \[ \frac{3y^2}{2}=-x^2+C \]
Step 3: Rearrange the equation. \[ x^2+\frac{3y^2}{2}=C \] Multiplying by \(2\), \[ 2x^2+3y^2=C_1 \] where \(C_1\) is a constant.
Step 4: Compare with standard conic. The equation \[ 2x^2+3y^2=C_1 \] contains positive coefficients of both \(x^2\) and \(y^2\). This represents an ellipse.
Step 5: Final answer. \[ \boxed{\text{Ellipse}} \]
To identify the family of curves represented by a differential equation, solve the differential equation and compare the obtained equation with standard forms of conics. The given equation is \[ 3y\frac{dy}{dx}+2x=0 \]
Step 1: Separate the variables. \[ 3y\frac{dy}{dx}=-2x \] Multiply both sides by \(dx\): \[ 3y\,dy=-2x\,dx \]
Step 2: Integrate both sides. \[ \int 3y\,dy=\int -2x\,dx \] \[ \frac{3y^2}{2}=-x^2+C \]
Step 3: Rearrange the equation. \[ x^2+\frac{3y^2}{2}=C \] Multiplying by \(2\), \[ 2x^2+3y^2=C_1 \] where \(C_1\) is a constant.
Step 4: Compare with standard conic. The equation \[ 2x^2+3y^2=C_1 \] contains positive coefficients of both \(x^2\) and \(y^2\). This represents an ellipse.
Step 5: Final answer. \[ \boxed{\text{Ellipse}} \]
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