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Alecsey [184]
3 years ago
13

Dy/dx = y/x , y(1) = −2

Mathematics
1 answer:
s2008m [1.1K]3 years ago
5 0

Start with

\dfrac{dy}{dx}=\dfrac{y}{x}

Separate the variables:

\dfrac{dy}{y} = \dfrac{dx}{x}

Integrate both parts:

\displaystyle \int \dfrac{dy}{y} = \int\dfrac{dx}{x}

Which implies

\log(y) = \log(x)+c

Solving for y:

y = e^{\log(x)+c} = e^{\log(x)}e^c=xe^c

Since e^c is itself a constant, let's rename it c_1.

Fix the additive constant imposing the condition:

y(1) = c_1\cdot 1 = -2\iff c_1=-2

So, the solution is

y(x) = -2x

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