Finding the general solution of the differential equation $y' = xy$

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Find the general solution of the differential equation: $y' = xy$.

Disagreeing with my friend on this, what's your take?

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2 Answers

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The general solution to this ODE is $y(x)=C e^\frac{x^2}{2}$. This solution can be found using separation of variables.

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$$\frac{dy}{dx}=-xy$$

$$\frac{1}{y}\frac{dy}{dx}=-x $$

$$\frac{1}{y}dy=-x dx \iff \int\frac{1}{y}=\int-x$$

$$\ln (y)+c=\frac{-(x^2)}{2} +c$$

$$2\ln(y)+c=-x^2+c$$

$$\ln y^2 = x^2 + c \iff y^2 = Ce^{x^2}\iff y = ce^{\frac{x^2}{2}}$$

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