Calculus, EXT1 EQ-Bank 28

Given the differential equation \(\dfrac{d y}{d x}=-\dfrac{x}{y e^{x^2}}\), determine the particular solution that passes through the point \((0,1 )\). (3 marks)

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\(y=e^{-\tfrac{x^2}{2}}\)

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\(\dfrac{d y}{d x}\)	\(=-\dfrac{x}{y e^{x^{2}}}\)
\(\displaystyle \int y\, d y\)	\(=-\displaystyle \int x e^{-x^2}\, d x\)
\(\dfrac{y^2}{2}\)	\(=\displaystyle \dfrac{1}{2} \int(-2 x) e^{-x^2}\, d x\)
\(y^2\)	\(=e^{-x^2}+c\)

\(\text{Given the solution passes through}\ (0,1):\)

\(1^2=e^0+c \ \ \Rightarrow \ \ c=0\)

\(y^2=e^{-x^2}\)

\(y=\left(e^{-x^2}\right)^{\tfrac{1}{2}}=e^{-\tfrac{x^2}{2}}\)