find the general form of f if f(x)= - 5f(x).\n\nf(x)=□

Explanation:

Step1: Recognize as ODE

We have a first - order ordinary differential equation $\frac{df(x)}{dx}=-5f(x)$.

Step2: Separate variables

Rewrite it as $\frac{df(x)}{f(x)}=-5dx$.

Step3: Integrate both sides

Integrating $\int\frac{df(x)}{f(x)}=\int - 5dx$. The left - hand side integral is $\ln|f(x)|$ and the right - hand side is $-5x + C$, so $\ln|f(x)|=-5x + C$.

Step4: Solve for f(x)

Exponentiate both sides: $f(x)=e^{-5x + C}=e^{C}e^{-5x}$. Let $A = e^{C}$, then $f(x)=Ae^{-5x}$, where $A$ is an arbitrary non - zero constant. When $A = 0$, $f(x)=0$ also satisfies the original ODE.

Answer:

$f(x)=Ae^{-5x}$, where $A$ is an arbitrary constant.