Goal: given \(\frac{dy}{dx}\propto y\), \(y(0)=y_0\), \(y(T)=y_T\) and \(T\ne0\), find \(y(x)\) in terms of \(y_0\), \(y_T\) and \(T\).
dxdydxdy∫y1dyln∣y∣yyy(x)∝y=ky=∫kdx=kx+C=exp(kx+C)=exp(C)exp(kx)=Cexp(kx)
y(x)y(0)y0y0y0=Cexp(kx)=Cexp(k(0))=Cexp(0)=C(1)=C
y(x)y(x)y(T)y0yTln(y0yT)T1ln(y0yT)=Cexp(kx)=y0exp(kx)=y0exp(kT)=exp(kT)=kT=k
y(x)y(x)y(x)=Cexp(kx)=y0exp(T1ln(y0yT)x)=y0(exp(ln(y0yT)))T1x
∴y(x)=y0(y0yT)Tx
#Calculus
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*
or_italics_
**bold**
or__bold__
paragraph 1
paragraph 2
[example link](https://brilliant.org)
> This is a quote
\(
...\)
or\[
...\]
to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
There are no comments in this discussion.