Thread: Seperable DE

Hey,

I've been having some trouble recreating the answer my teacher has for the following question:

Find $\displaystyle y(x)$ given $\displaystyle \frac{dy}{dx} = x^3y$, where $\displaystyle y(1) = 2$.

So I did the following:

$\displaystyle \int \frac{dy}{y} = \int x^3dx \longrightarrow \ln{y} = \frac{x^4}{4} + C$

$\displaystyle y = e^{\frac{x^4}{4} + C} \ \ or \ \ y = Ce^{\frac{x^4}{4}}$

Where C is some constant to be solved for.

Anyway I could solve for C. But what is getting me is that my prof's final answer is:

$\displaystyle y = 2e^{\frac{x^4}{4} - \frac{1}{4}}$

Which would mean there was two constants...but I would need more initial conditions for that.

Originally Posted by TrevorP

Hey,

I've been having some trouble recreating the answer my teacher has for the following question:

Find $\displaystyle y(x)$ given $\displaystyle \frac{dy}{dx} = x^3y$, where $\displaystyle y(1) = 2$.

So I did the following:

$\displaystyle \int \frac{dy}{y} = \int x^3dx \longrightarrow \ln{y} = \frac{x^4}{4} + C$

$\displaystyle y = e^{\frac{x^4}{4} + C} \ \ or \ \ y = Ce^{\frac{x^4}{4}}$

Where C is some constant to be solved for.

Anyway I could solve for C. But what is getting me is that my prof's final answer is:

$\displaystyle y = 2e^{\frac{x^4}{4} - \frac{1}{4}}$

Which would mean there was two constants...but I would need more initial conditions for that.

You have that $\displaystyle 2=Ce^{\frac{1^4}{4}}$ dividing each side we get $\displaystyle c=\frac{2}{e^{\frac{1}{4}}}\Rightarrow2e^{\frac{-1}{4}}$ see it now?

Hello, Trevor!

You're both correct . . .

$\displaystyle y \:= \:e^{\frac{x^4}{4} + C}\:\text{ or } \:y \:= \:Ce^{\frac{x^4}{4}}$ .where $\displaystyle C$ is some constant to be solved for.

I could solve for C. . . . . Did you?

$\displaystyle \text{My prof's final answer is: }\;y \:= \:2\,e^{\frac{x^4}{4} - \frac{1}{4}}$

Since $\displaystyle y(1) = 2$, we have: .$\displaystyle Ce^{\frac{1}{4}} \:=\:2\quad\Rightarrow\quad C \:=\:2e^{\text{-}\frac{1}{4}}$


Then the function becomes: .$\displaystyle y \;=\;2\underbrace{e^{\text{-}\frac{1}{4}}\cdot e^{\frac{x^2}{4}}}_{\text{combine}} \;=\;2\,e^{ \frac{x^2}{4}-\frac{1}{4}} $ . . . see?

Yeah I did..but something just wasn't clicking for some reason. I just thought that since two numbers were introduced there should be two variables.

But now it makes sense.