Can someone Check my work for a differential equation work please? It ask to prove that the function is a solution to the differential equation. Give me a min to use the function builder to input the formulas please.

Question

anonymous · Answer

$$\frac{dy}{dx}=ay$$ 

assume a is a coefficient

anonymous · Answer

$$\frac{1}{ay}dy=dx$$

anonymous · Answer

$$\int\limits(\frac{1}{ay})dy=\int\limits dx$$

anonymous · Answer

$$\log(ay)=x+c$$

anonymous · Answer

$$y=\frac{e^{x+c}}{a} $$

anonymous · Answer

$$\lim_{y \rightarrow o}y=\lim_{x \rightarrow 0}\frac{e^{x+c}}{a}$$

anonymous · Answer

0=0

anonymous · Answer

So that proves that it's the solution right?

turingtest · Answer

no, because $$\lim_{x \rightarrow 0}\frac{e^{x+c}}{a}=\frac{e^c}a$$and we don't know what c is, and even if c was 0, that would give$$\lim_{x \rightarrow 0}\frac{e^{x+c}}{a}=\frac{e^0}a=\frac1a$$

turingtest · Answer

what is the function that you are trying to prove is a solution?

turingtest · Answer

I may not be online by the time you reply, but if you have a function for y, I suggest you simply plug it in to the differential equation above and make sure it comes out the same on both sides.

turingtest · Answer

that would prove it as a solution to the DE

anonymous · Answer

What makes this a pain is that I don't have the book so I'm using pictures that a class mate took for me. Let me post the picture see if you have any better luck understanding it than I do. It's problem number 1 on this page i'm about to upload.

anonymous · Answer

attachment

anonymous · Answer

oh crap I didn't see the functions on the right of the diff eq

loser66 · Answer

$$\frac{dy}{dx}= ay $$ solution is $y = e^{ax}$ right?