Find the Solution to the Differential Equation: $$~\frac{dy}{dt}(t) - ay(t) = 0 , ~\text{with} ~~~~~~~y(1) = \pi~$$ With $~a = 1,~~~ y(2) = 8.54~$
Attempt: I tried multiple guesses to what $~y(t)~$ would be $~( \pi ~ t, ~\pi ^t )~$ and then tried to solve for $~a~$, but I cannot find a way to balance the equation.
Given that $$~\frac{dy}{dt}(t) - ay(t) = 0 \tag1,$$
The above equation is a linear differential equation of first order.
Integrating factor (I.F.) is $~e^{-\int a ~dt}=e^{-at}~$
Multiplying the given equation $(1)$ with the I.F. and then integrating we have $$\int\left(e^{-at}~\frac{dy}{dt}(t) - a~e^{-at}~y(t)\right)dt = 0$$ $$\implies\int d\left(y~e^{-at}\right)=0$$ $$\implies y~e^{-at}=c\qquad \text{where $~c~$is constant.}\tag2$$ Since $~y(1) = \pi\implies~\pi~e^{-a}~=c$
From $(2)$, we have $$y~e^{-at}=\pi~e^{-a}\implies y=\pi~e^{a(t-1)}$$ When $~a=1,~x=2~$, $~y(2)=\pi\cdot e~=8.5397\equiv 8.54$
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We know $~\pi= 3.14159~$ and $~e=2.71828~$