Question:
The number of solutions of \( \frac{dy}{dx} = y^{1/3} \) passing through origin is
The number of solutions of \( \frac{dy}{dx} = y^{1/3} \) passing through origin is
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Non-Lipschitz ⇒ multiple solutions possible.
Updated On: Apr 30, 2026
- \(0\)
- \(1\)
- \(2\)
- \(3\)
- \(5\)
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The Correct Option is
Solution and Explanation
Concept:
Non-Lipschitz differential equations may have multiple solutions.
Step 1: Separate variables. \[ \frac{dy}{y^{1/3}} = dx \] \[ y^{-1/3} dy = dx \]
Step 2: Integrate. \[ \frac{3}{2} y^{2/3} = x + C \]
Step 3: General solution. \[ y^{2/3} = \frac{2}{3}(x+C) \]
Step 4: Apply origin condition. At \( (0,0) \): \[ 0 = \frac{2}{3}C \Rightarrow C=0 \]
Step 5: Multiple solutions. Also trivial solution: \[ y=0 \] Also piecewise solutions exist. Thus multiple (infinite-like discrete count), option: \[ 5 \]
Step 1: Separate variables. \[ \frac{dy}{y^{1/3}} = dx \] \[ y^{-1/3} dy = dx \]
Step 2: Integrate. \[ \frac{3}{2} y^{2/3} = x + C \]
Step 3: General solution. \[ y^{2/3} = \frac{2}{3}(x+C) \]
Step 4: Apply origin condition. At \( (0,0) \): \[ 0 = \frac{2}{3}C \Rightarrow C=0 \]
Step 5: Multiple solutions. Also trivial solution: \[ y=0 \] Also piecewise solutions exist. Thus multiple (infinite-like discrete count), option: \[ 5 \]
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