Question:
The solution of the differential equation \( y'(y^2 - x) = y \) is
The solution of the differential equation \( y'(y^2 - x) = y \) is
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When expression resembles total differential, try constructing \(d(f(x,y))\) form.
Updated On: May 8, 2026
- \( y^3 - 3xy = C \)
- \( y^3 + 3xy = C \)
- \( x^3 - 3xy = C \)
- \( y^3 - xy = C \)
- \( x^3 - xy = C \)
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The Correct Option is A
Solution and Explanation
Concept:
Convert into separable or exact form:
\[
\frac{dy}{dx} = \frac{y}{y^2 - x}
\]
Step 1: Rearrange
\[ (y^2 - x)dy = y dx \]
Step 2: Rewrite
\[ y^2 dy - x dy = y dx \] Rearrange: \[ y^2 dy - y dx - x dy = 0 \]
Step 3: Identify exact differential
Try: \[ d(y^3 - 3xy) \] \[ = 3y^2 dy - 3x dy - 3y dx \] Divide by 3: \[ = y^2 dy - x dy - y dx \] Matches given expression.
Step 4: Integrate
\[ d(y^3 - 3xy) = 0 \] \[ y^3 - 3xy = C \] Final Answer: \[ \boxed{y^3 - 3xy = C} \]
Step 1: Rearrange
\[ (y^2 - x)dy = y dx \]
Step 2: Rewrite
\[ y^2 dy - x dy = y dx \] Rearrange: \[ y^2 dy - y dx - x dy = 0 \]
Step 3: Identify exact differential
Try: \[ d(y^3 - 3xy) \] \[ = 3y^2 dy - 3x dy - 3y dx \] Divide by 3: \[ = y^2 dy - x dy - y dx \] Matches given expression.
Step 4: Integrate
\[ d(y^3 - 3xy) = 0 \] \[ y^3 - 3xy = C \] Final Answer: \[ \boxed{y^3 - 3xy = C} \]
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