### Author Topic: Q1 problem 1 (day section)  (Read 1825 times)

#### Victor Ivrii

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##### Q1 problem 1 (day section)
« on: September 29, 2014, 02:21:19 AM »
Please post problem and solution

#### Rhoda Lam

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##### Re: Q1 problem 1 (day section)
« Reply #1 on: October 04, 2014, 09:42:22 PM »
2.1 p. 40, #18
\begin{gather}
ty' + 2y = \sin t,\\
y(Ï€/2) = 1
\end{gather}
First, change the equation by dividing every term by $t$.

y' + (2/t)y = (\sin (t))/t

Then, solve for the integrating factor.

Î¼ = e^{\int (2/t)\,dt} = e^{2\ln(t)\,} = t^2

Multiply every term in equation by Î¼.

t^2y' + 2ty = t\sin(t)

Integrate both sides of the equation.
\begin{gather}
\int[t^2y]' = \int{t\sin(t)}dt\implies
t^2y = âˆ’t\cos(t) âˆ’ \int{-\cos(t)}dt= âˆ’t\cos(t) + \sin(t) + C\implies y = (âˆ’t \cos(t) +\sin (t) + C)/t^2
\end{gather}
Substitute the initial value to solve for C.
\begin{gather}
C = (Ï€/2)^2-1
\end{gather}
Therefore, the solution is:
\begin{gather}
y = (âˆ’t \cos(t) +\sin (t) + (Ï€/2)^2-1)/t^2
\end{gather}
« Last Edit: October 05, 2014, 02:25:04 AM by Victor Ivrii »