MHT CET · Maths · Differential Equations
The equation of the curve whose slope at any point is equal to \(2 x y\) and which passes through the point \((0,1)\) is
- A \(\cdot \log y=x^{2}\)
- B \(\quad \log y=\frac{1}{x}\)
- C \(\frac{1}{y}=x\)
- D \(\log y=x\)
Answer & Solution
Correct Answer
(A) \(\cdot \log y=x^{2}\)
Step-by-step Solution
Detailed explanation
We have \(\frac{d y}{d x}=2 x y\)
\(\therefore \int \frac{\mathrm{dy}}{\mathrm{y}}=\int 2 \mathrm{x} \mathrm{dx}\)
\(\Rightarrow \log y=\frac{2 x^{2}}{2}+C\)
\(\log y=x^{2}+C\) and curve passes through point \((0,1)\).
At \(x=0, y=1\), we get \(C=0\)
\(\therefore \log y=x^{2}\)
\(\therefore \int \frac{\mathrm{dy}}{\mathrm{y}}=\int 2 \mathrm{x} \mathrm{dx}\)
\(\Rightarrow \log y=\frac{2 x^{2}}{2}+C\)
\(\log y=x^{2}+C\) and curve passes through point \((0,1)\).
At \(x=0, y=1\), we get \(C=0\)
\(\therefore \log y=x^{2}\)
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