JEE Mains · Maths · STD 12 - 9. differential equations
The solution of the differential equation \(\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0\) is (where \(C\) is a constant of integration.)
- A \(x-2 \log _{e}(y+3 x)=C\)
- B \(x-\log _{e}(y+3 x)=C\)
- C \(x-\frac{1}{2}\left(\log _{e}(y+3 x)\right)^{2}=C\)
- D \(y+3 x-\frac{1}{2}\left(\log _{e} x\right)^{2}=C\)
Answer & Solution
Correct Answer
(C) \(x-\frac{1}{2}\left(\log _{e}(y+3 x)\right)^{2}=C\)
Step-by-step Solution
Detailed explanation
\(\ell n ( y +3 x )= z (\) let \()\) \(\frac{1}{y+3 x} \cdot\left(\frac{d y}{d x}+3\right)=\frac{d z}{d x}\) \(\frac{d y}{d x}+3=\frac{y+3 x}{\ell n(y+3 x)} \quad(\) given \()\) \(\frac{ dz }{ dx }=\frac{1}{ z }\) \(\Rightarrow z dz = dx \Rightarrow \frac{ z ^{2}}{2}= x + C\)…
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