JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be solution of the differential equation \(\log _{e}\left(\frac{d y}{d x}\right)=3 x+4 y\), with \(y(0)=0\). If \(y\left(-\frac{2}{3} \log _{e} 2\right)=\alpha \log _{e} 2\), then the value of \(\alpha\) is equal to:
- A \(-\frac{1}{2}\)
- B \(-\frac{1}{4}\)
- C \(2\)
- D \(\frac{1}{4}\)
Answer & Solution
Correct Answer
(B) \(-\frac{1}{4}\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}=e^{3 x} \cdot e^{4 y} \Rightarrow \int e^{-4 y} d y=\int e^{3 x} d x\) \(\frac{e^{-4 y}}{-4}=\frac{e^{3 x}}{3}+C \Rightarrow-\frac{1}{4}-\frac{1}{3}=C \Rightarrow C=-\frac{7}{12}\)…
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