1. True/False Proof

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

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

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

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

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

$\displaystyle = x^2 + C$.

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

Now since $\displaystyle y = x^2 + c$

$\displaystyle x^2 = y - c$

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

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

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

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

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

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

So the statement is true.