CUET · MATHS · PYQ PAPER 2025
The solution of the differential equation \(\log _e\left(\frac{d y}{d x}\right)=5 x+2 y\) is given by
- A \(5 e^{5 x}+2 e^{-2 y}+C=0: C\) is an arbitrary constant
- B \(2 e^{5 x}+5 e^{-2 y}+C=0: C\) is an arbitrary constant
- C \(2 e^{-5 x}+5 e^{2 y}+C=0: C\) is an arbitrary constant
- D \(5 e^{-5 x}+2 e^{2 y}+C=0: C\) is an arbitrary constant
Answer & Solution
Correct Answer
(B) \(2 e^{5 x}+5 e^{-2 y}+C=0: C\) is an arbitrary constant
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x} = e^{5 x+2 y} = e^{5 x} e^{2 y}\) \(e^{-2 y} d y = e^{5 x} d x\) \(\int e^{-2 y} d y = \int e^{5 x} d x\) \(\frac{e^{-2 y}}{-2} = \frac{e^{5 x}}{5} + C'\) \(5 e^{-2 y} = -2 e^{5 x} - 10C'\) \(2 e^{5 x} + 5 e^{-2 y} + C = 0\)
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