# Differential equation solution through integration

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.

• $e^{A+B}=e^A·e^B$. Nov 13, 2021 at 6:52
• ...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.) Nov 13, 2021 at 14:39

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 $$x=f(t)$$.