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If I have the following differential equation

$$ dx/dt=x $$

The ‘normal’ way of solving a differential equation is with separation of variables

$$ dx/x=dt $$

Integrating both sides you get

$$ ln(x)=t + C $$

Raising to the power of e

$$ x(t)=e^t+e^C $$

Why isn’t this equally as valid

$$ dx/dt=x $$

Integrate both sides with respect to t

$$ x=xt+C $$

Which obviously doesn’t match. But also another possible solution would be

$$ x $$

Taking the derivative you get

$$ dx/dt = x $$

What am I missing? Is this just a matter of me misunderstanding the syntax? The other thing I notice is with separation of variables the left side is integrated with respect to x but right hand with respect to y.

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    $\begingroup$ $e^{A+B}=e^A·e^B$. $\endgroup$
    – Lutz Lehmann
    Nov 13, 2021 at 6:52
  • $\begingroup$ ...and besides that algebra error, you also haven't taken the possibility $x \le 0$ into account. You need to consider $x=0$ separately, and include absolute values in $\ln|x| = t+C$. (A simpler way which avoids this hassle is to use the method of integrating factors instead.) $\endgroup$
    – Hans Lundmark
    Nov 13, 2021 at 14:39

1 Answer 1

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Because$$x'(t)=x(t)\Leftrightarrow\int x'(t)dt=\int x(t)dt\underset{\text{generally}}{\neq} x(t)t+C$$ You mustn't forget that an expression like $dx/dt$ means that $x=f(t)$.

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