MAT244-2014F > Quiz 1

Q1 problem 2 (Night sections)

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**Victor Ivrii**:

Please post solution

2.6 p. 102, # 25 Solve the initial value problem

\begin{equation*}

y'-2y=e^{2t},\qquad y(0)=2.

\end{equation*}

**Chang Peng (Eddie) Liu**:

Firstly, I'd like to apologize for the really ugly notations and equations.. there is the solution! I rewrote it--V.I.

Find the solution of the given initial value problem.

\begin{gather}

y' - 2y = e^{2t},\label{A}\\

y(0) = 2.\label{B}

\end{gather}

First, we need to find the integrating factor, which is $I = e^{\int -2\,dt}= e^{-2t}$

so we multiple the entire equation by I, thus, giving us

$e^{-2t} y' - 2e^{-2t} y = 1$.

We see that the left side of the equation can be rewritten as $[e^{-2t}y]'$

and we see that the right side of the equation is in fact 1

therefore: $[e^{-2t} y]' = 1$

we now take the integral of both sides, giving us: $e^{-2t} y = t + C $ where $C$ is a constant. To find $C$, we substitute $y = 2$ when $t$ = 0 (this information is given in the question). We find out that $C = 2$.

rearranging the formula, we come to the solution as

\begin{equation*}

y = (t+2) e^{2t}.

\end{equation*}

**Weiyang Guo**:

haha, I saw the answer was wrong this afternoon (without the minus sign on 2t) and I thought I should go home and reply about this. But it's fixed so quickly. :)

**Bowen Cheng**:

I dont use intergrating factor for this problem, instead I use an alternative approach

y' - 2y = e ^(2t) ---------(1)

--> z' -2z = 0 (assume the R.H.E. =0), then dz\z=2dt --> âˆ«dz/z =2dt --> ln |z| = 2t+C1

-->|z| = C1e^(2t) --> z = Ce^(2t), C = Â±C1

-->now let C = C(t), such that y= Ce^(2t)--------------(2)

plug (2) into (1), we get C'e^(2t) + 2Ce^(2t)- 2Ce^(2t) = e^(2t) --->C'e^(2t) = e^(2t)

--->C' = 1 ---> C = t + C2 ---> y = Ce^(2t) = ( t + C2)e^(2t) ----(3)

now plug in (0,2) into (3) -->2 =C2 -->therefore y = ( t + 2)e^(2t)

**Victor Ivrii**:

--- Quote from: ID1000594047 on September 26, 2014, 08:01:30 PM ---I dont use intergrating factor for this problem, instead I use an alternative approach

--- End quote ---

It is difficult to decrypt. Please rewrite (quote one of the previous posts to see how to display math properly), check spelling and fix your name

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