I am stuck on how to solve the following, i don't weather my arithmetic is wrong or what i am doing. So could someone please do each step.
Find the particular solution to
a) dh/dt=2h(3-h) where y=1 and t=0
b) dy/dx= e^y +1/e^y where y=0 when x=0.
I am stuck on how to solve the following, i don't weather my arithmetic is wrong or what i am doing. So could someone please do each step.
Find the particular solution to
a) dh/dt=2h(3-h) where y=1 and t=0
b) dy/dx= e^y +1/e^y where y=0 when x=0.
First of all try to find the general solution of the DE (it's a separable one), for example the first one:
$\displaystyle \frac{dh}{dt}=2h(3-h)$
Calculating general solution, first rewrite the DE:
$\displaystyle \frac{dh}{2h(3-h)}=dt$
Take the integral of both sides:
$\displaystyle \int \frac{dh}{2h(3-h)}=\int dt$
If you split the integrand in partial fractions (in LHS) you'll get:
$\displaystyle \frac{1}{6}\int \frac{dh}{h} - \frac{1}{6}\int \frac{dh}{h-3}=tt+C$
$\displaystyle =\frac{1}{6}\ln|h|-\frac{1}{6}\ln|h-3|=t+C$
$\displaystyle \Leftrightarrow \ln\left|\frac{h}{h-3}\right|=6t+6C$
$\displaystyle \Leftrightarrow e^{6t+6C}=\left|\frac{h}{h-3}\right|$
Now try to continue and try to get an expression $\displaystyle h(t)=...$ and afterwards substitute $\displaystyle h=1,t=0$ to find $\displaystyle C$.
(Note: $\displaystyle \ln$ is the natural logarithm)
Hiya Johnsy, is the second one $\displaystyle \displaystyle \frac{dy}{dx} = e^y + \frac{1}{e^y}$ or $\displaystyle \displaystyle \frac{dy}{dx} = \frac{e^y + 1}{e^y}$?
For the first...
$\displaystyle \displaystyle \begin{align*} \frac{dh}{dt} &= 2h(3 - h) \\ \frac{dt}{dh} &= \frac{1}{2h(3-h)} \\ t &= \int{\frac{1}{2h(3 -h)}\,dh} \end{align*}$
You will now need to apply partial fractions.
OK, so
$\displaystyle \displaystyle \begin{align*} \frac{dy}{dx} &= \frac{e^y + 1}{e^y} \\ \frac{dx}{dy} &= \frac{e^y}{e^y + 1} \\ x &= \int{\frac{e^y}{e^y + 1}\,dy} \end{align*}$
Now make the substitution $\displaystyle \displaystyle u = e^y + 1 \implies du = e^y\,dy$ and the integral becomes
$\displaystyle \displaystyle x = \int{\frac{1}{u}\,du}$.
I'm sure you can go from here
... so that the DE is...
$\displaystyle y^{'}= 1+e^{-y}\ ,\ y(0)=0$ (1)
In (1) the variables are separable and the solution is found with the standard approach...
$\displaystyle \int \frac{d y}{1+e^{-y}} = \int dx \implies y+ \ln (1+e^{-y}) = x + c$ (2)
... and the 'initial condition' means that is $\displaystyle c=0$. All right?... yes, of course, but it is interesting to spend a little of time about the details of the integration performed in (2). The integral is...
$\displaystyle \int \frac{d y}{1+e^{-y}} = \int dy - \int \frac{e^{-y}}{1+e^{-y}}\ dy= y+\ln (1+e^{-y}) + c$ (3)
The integral (3) is not 'trascendental' but may be that someone prefers to use Wolfram because 'it saves time'... well!... please observe the result supplied by Wolfram ...
Wolfram Mathematica Online Integrator
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Of course different methods can give the same result... in this case the 'Year 12 Specialist Mathematics method' gives the solution...
$\displaystyle x= \ln (1+e^{y}) + c$ (1)
... and the 'chisigma method' gives the solution...
$\displaystyle x= y + \ln (1+e^{-y}) + c = y + \ln (1+e^{-y}) + c$ (2)
... but considering the 'identity'...
$\displaystyle \ln (1+e^{y}) = \ln e^{y} + \ln (1+e^{-y})= y + \ln (1+e^{-y}) $ (3)
... it is obvious that (1) and (2) are exactly the same expression. Scope [a little polemical may be...] of my post was to do some considerations about the 'reliability' of some 'very popular tools' ...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$