Find the general expression from the antiderivative

Question

I am having trouble computing the original function.

Question states:

Let $f$ be a differentiable, positive function, such that

$$f'(x)=x*f(x)$$

for all real numbers x.

A) Find the general expression for the function $f$.

The stated derivative automatically rings chain rule for me.

Trying to take the indefinite integral of the derivative didn't result in anything.

The function that is closest to the intended result that I can think of would be $e^x$ but that returns $f(x)*x'$.

Any help would be appreciated, specifically with how I should go about solving these types of problems. Thank you!

Answer 1

$$(d/dx) \ln f(x) = \frac{f'(x)}{f(x)} = x.$$

Therefore $\ln f(x) = \frac{1}{2}x^2 + C$, so $f(x) = e^{\frac{1}{2}x^2 + C} = De^{\frac{1}{2}x^2}$, where $D$ is an arbitrary positive constant.

Answer 2

$$ f' = x f \iff \int \frac{ df}{f} = \int x dx \implies \ln |f| = x^2 + C \implies f(x) = \tilde C e^{x^2} $$ with $C \in \mathbb{R}$