1. ## True/False Proof

True or false: If $f(x)$ is a solution to $\frac{dy}{dx}=2x$, then $f'(x)$ is a solution to $\frac{dy}{dx}=2y$. Justify your answer.

2. Originally Posted by jangalinn
True or false: If $f(x)$ is a solution to $\frac{dy}{dx}=2x$, then $f'(x)$ is a solution to $\frac{dy}{dx}=2y$. Justify your answer.

I assume you're letting $y = f(x)$ and $\frac{dy}{dx} = f'(x)$.

$\frac{dy}{dx} = 2x$

$y = \int{2x\,dx}$

$= x^2 + C$.

So $y = f(x) = x^2 + c$.

Now since $y = x^2 + c$

$x^2 = y - c$

$x = (y - c)^{\frac{1}{2}}$

$\frac{dx}{dy} = \frac{1}{2}(y - c)^{-\frac{1}{2}}$

$\frac{dx}{dy} = \frac{1}{2(y - c)^{\frac{1}{2}}}$

$\frac{dy}{dx} = 2(y - c)$

$\frac{dy}{dx} = 2y - 2c$

$f'(x) = 2y + C$ where $C = -2c$.

So the statement is true.