Initial-value condition, undefined constant

Question

I have to find the exact solution, and then plot it against Euler's method, but the results I'm getting from using my formula don't seem right. Starting out with the equation $$T'=\frac{-1}4(T-20)$$ with $T(0)=200$, c seems to be undefined? I've integrated and applied the initial conditions $T=200$ and $t=0$: $$-4\ln(-180)=c$$ which makes c undefined. Where am I going wrong? I've put it through Symbolab and it also says it's undefined, I've tried rearranging it before applying the initial conditions, getting $$T=20-e^{-4/t}+c$$ which gives $c=181$, and $$T=201-e^{-4/t}$$ This doesn't satisfy the initial condition as you can't divide by $0$, and gives me T values that barely change.

Answer 1

Remember that

$\int \frac{4}{20-y} \, dy=-4 \log |y-20|$

so when you solve your equation $\dfrac{4\,\text{dy}}{y-20}=\text{dt}$

you get

$-4 \log |y-20|+C=t$

with initial conditions

$-4 \log 180+C=0\to C= 4\log 180$

so you have

$-4\log|y-20|+4\log 180=t$

$-4(\log\dfrac{|y-20|}{180}=t$

$\log\dfrac{|y-20|}{180}=-\dfrac{t}{4}$

$\dfrac{|y-20|}{180}=e^{-\frac{t}{4}}$

$y=180 e^{-\frac{t}{4}}+20$