Question

find the general form of f if f(x)= - 2f(x).
f(x)= square

Explanation:

Step1: Recognize as ODE

We have a first - order ordinary differential equation $\frac{df(x)}{dx}=-2f(x)$.

Step2: Separate variables

Rewrite it as $\frac{df(x)}{f(x)}=-2dx$.

Step3: Integrate both sides

Integrate $\int\frac{df(x)}{f(x)}=\int - 2dx$. The left - hand side integral is $\ln|f(x)|$ and the right - hand side is $-2x + C$, so $\ln|f(x)|=-2x + C$.

Step4: Solve for f(x)

Exponentiate both sides: $|f(x)| = e^{-2x + C}=e^{C}e^{-2x}$. Let $A = e^{C}$ (where $A
eq0$), then $f(x)=Ae^{-2x}$. Also, $f(x) = 0$ is a solution (when $A = 0$). So the general solution is $f(x)=Ae^{-2x}$, where $A$ is an arbitrary constant.

Answer:

$f(x)=Ae^{-2x}$