MHT CET · Maths · Differential Equations
The solution of the differential equation \(y(1+\log x) \frac{d x}{d y}-x \log x=0\) is
- A \(x \log x=y+c\)
- B \(x \log x=y c\)
- C \(y(1+\log x)=c\)
- D \(\log x-y=c\)
Answer & Solution
Correct Answer
(B) \(x \log x=y c\)
Step-by-step Solution
Detailed explanation
Given differential equation is
\(
y(1+\log x) \frac{d x}{d y}-x \log x=0
\)
\(\Rightarrow \int \frac{1+\log x}{x \log x} d x=\int \frac{d y}{y} \)
\( \Rightarrow \int \frac{1}{x \log x} d x+\int \frac{1}{x} d x=\int \frac{1}{y} d y \)
\( \Rightarrow \log (\log x)+\log x=\log y+\log c \)
\( \Rightarrow x \log x=y c\)
\(
y(1+\log x) \frac{d x}{d y}-x \log x=0
\)
\(\Rightarrow \int \frac{1+\log x}{x \log x} d x=\int \frac{d y}{y} \)
\( \Rightarrow \int \frac{1}{x \log x} d x+\int \frac{1}{x} d x=\int \frac{1}{y} d y \)
\( \Rightarrow \log (\log x)+\log x=\log y+\log c \)
\( \Rightarrow x \log x=y c\)
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