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vlada-n [284]
1 year ago
7

05* Find, for y> 0, the general solution of the differential equation dy/dx=xy.

Mathematics
1 answer:
Alla [95]1 year ago
8 0

The ODE is separable.

\dfrac{dy}{dx} = xy \iff \dfrac{dy}y = x\,dx

Integrate both sides to get

\displaystyle \int\frac{dy}y = \int x\,dx

\boxed{\ln|y| = \dfrac12 x^2 + C}

But notice that replacing the constant C with -C doesn't affect the solution, since its derivative would recover the same ODE as before.

\ln|y| = \dfrac12 x^2 - C \implies \dfrac1y \dfrac{dy}{dx} = x \implies \dfrac{dy}{dx} = xy

so either of the first two answers are technically correct.

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