# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Quiz-1 => Topic started by: Victor Ivrii on September 28, 2018, 03:36:35 PM

Title: Q1: TUT 0701
Post by: Victor Ivrii on September 28, 2018, 03:36:35 PM
Find the solution of the given initial value problem
\begin{equation*}
t^3y' + 4t^2y = e^{-t},\qquad  y(-1)=0,\qquad t<0.
\end{equation*}
Title: Re: Q1: TUT 0701
Post by: Tzu-Ching Yen on September 28, 2018, 06:00:02 PM
First divide by $t^3$ on both side of the equation, we get
$$y' + \frac{4}{t}y = \frac{e^{-t}}{t^3}$$
Using the method of integrating factor we have equation for $u(t)$
$$u(t) = e^{\int \frac{4}{t}dt} = e^{4\ln(t) + c} = t^4$$
where constant $c$ is arbitrary, it's chosen to be 0 here. Then
$$\bigl(y u(t)\bigr)' = u(t)\frac{e^{-t}}{t^3}$$
rearranging gives equation
$$y = \frac{1}{u(t)}\int u(t)\frac{e^{-t}}{t^3}$$
substitute in $u(t) = t^4$
$$y = \frac{1}{t^4}\int te^{-t}$$
use integration by parts
$$y = -\frac{e^{-t}}{t^3} - \frac{e^{-t}}{t^4} + \frac{c_1}{t^4}$$
to check $c_1$, plug in condition $y(-1) = 0$
$$y(-1) = e - e + c_1 = c_1= 0$$
Plug in $c_1 = 0$ gets
$$y = -\frac{e^{-t}}{t^3} - \frac{e^{-t}}{t^4}$$
Title: Re: Q1: TUT 0701
Post by: Nick Callow on September 28, 2018, 06:00:48 PM
My solution to this quiz can be found on this attachment.
Title: Re: Q1: TUT 0701
Post by: Victor Ivrii on September 29, 2018, 03:27:37 PM
Nick, still typed in forum solution is better then typed externally (but I applaud your typesetting skills)
The text passage contains a mathematical question and its solution. The question is about solving an initial value problem for a differential equation, and the solution is provided in detail by a user named Tzu-Ching Yen.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Quiz-1 => Topic started by: Victor Ivrii on September 28, 2018, 03:36:35 PM

Title: Q1: TUT 0701
Post by: Victor Ivrii on September 28, 2018, 03:36:35 PM
Find the solution of the given initial value problem
\begin{equation*}
t^3y' + 4t^2y = e^{-t},\qquad  y(-1)=0,\qquad t<0.
\end{equation*}
Title: Re: Q1: TUT 0701
Post by: Tzu-Ching Yen on September 28, 2018, 06:00:02 PM
First divide by $t^3$ on both side of the equation, we get
$$y' + \frac{4}{t}y = \frac{e^{-t}}{t^3}$$
Using the method of integrating factor we have equation for $u(t)$
$$u(t) = e^{\int \frac{4}{t}dt} = e^{4\ln(t) + c} = t^4$$
where constant $c$ is arbitrary, it's chosen to be 0 here. Then
$$\bigl(y u(t)\bigr)' = u(t)\frac{e^{-t}}{t^3}$$
rearranging gives equation
$$y = \frac{1}{u(t)}\int u(t)\frac{e^{-t}}{t^3}$$
substitute in $u(t) = t^4$
$$y = \frac{1}{t^4}\int te^{-t}$$
use integration by parts
$$y = -\frac{e^{-t}}{t^3} - \frac{e^{-t}}{t^4} + \frac{c_1}{t^4}$$
to check $c_1$, plug in condition $y(-1) = 0$
$$y(-1) = e - e + c_1 = c_1= 0$$
Plug in $c_1 = 0$ gets
$$y = -\frac{e^{-t}}{t^3} - \frac{e^{-t}}{t^4}$$
Title: Re: Q1: TUT 0701
Post by: Nick Callow on September 28, 2018, 06:00:48 PM
My solution to this quiz can be found on this attachment.
Title: Re: Q1: TUT 0701
Post by: Victor Ivrii on September 29, 2018, 03:27:37 PM
Nick, still typed in forum solution is better then typed externally (but I applaud your typesetting skills)
user
The text passage contains a mathematical question and its solution. The question is about solving an initial value problem for a differential equation, and the solution is provided in detail by a user named Tzu-Ching Yen.

TYPE: SINGLE_QUESTION_WITH_ANSWER