LatihanODE_01.lyx

#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\use_minted 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation skip
\defskip smallskip
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Standard
Kerjakan dengan menggunakan Jupyter Notebook.
 Lengkapi jawaban Anda dengan penjelasan singkat mengenai metode yang digunakan,
 hasil yang diperoleh, dan visualisasi.
\end_layout

\begin_layout Standard
\begin_inset CommandInset line
LatexCommand rule
offset "0.5ex"
width "100col%"
height "1pt"

\end_inset


\end_layout

\begin_layout Standard
(Chapra Latihan 25.2) Cari solusi dari persamaan diferensial berikut:
\begin_inset Formula 
\[
\frac{\mathrm{d}y}{\mathrm{d}t}=(1+4t)\sqrt{y}
\]

\end_inset

dengan ukuran langkah 0.25 dengan syarat awal 
\begin_inset Formula $y(t=0)=1$
\end_inset

 dengan menggunakan metode berikut:
\end_layout

\begin_layout Itemize
Metode Euler
\end_layout

\begin_layout Itemize
Metode Heun
\end_layout

\begin_layout Itemize
Metode Runge-Kutta orde-4
\end_layout

\begin_layout Standard
\begin_inset CommandInset line
LatexCommand rule
offset "0.5ex"
width "100col%"
height "1pt"

\end_inset


\end_layout

\begin_layout Standard
(Chapra Latihan 25.18) Selesaikan persamaan diferensial orde-2 berikut ini:
\begin_inset Formula 
\[
\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+5x\frac{\mathrm{d}x}{\mathrm{d}t}+(x+7)\sin(\omega t)=0
\]

\end_inset

dengan syarat awal:
\begin_inset Formula 
\[
\frac{\mathrm{d}x}{\mathrm{d}t}(0)=1.5
\]

\end_inset


\begin_inset Formula 
\[
x(0)=6
\]

\end_inset

dengan 
\begin_inset Formula $\omega=1$
\end_inset

.
 Selesaikan untuk rentang waktu 
\begin_inset Formula $t=0$
\end_inset

 sampai 
\begin_inset Formula $t=15$
\end_inset

 dan buat plot dari hasil yang diperoleh.4
\end_layout

\begin_layout Standard
Misalkan:
\begin_inset Formula 
\[
u_{1}(t)=x(t)
\]

\end_inset


\begin_inset Formula 
\[
u_{2}(t)=\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\frac{\mathrm{d}u_{1}(t)}{\mathrm{d}t}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\frac{\mathrm{d}u_{2}(t)}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\frac{\mathrm{d}^{2}x(t)}{\mathrm{d}t^{2}}=-5x\frac{\mathrm{d}x}{\mathrm{d}t}-(x+7)\sin(\omega t)
\]

\end_inset


\end_layout

\begin_layout Standard
——————————
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\frac{\mathrm{d}u_{1}(t)}{\mathrm{d}t}=u_{2}(t)
\]

\end_inset


\begin_inset Formula 
\[
\frac{\mathrm{d}u_{2}(t)}{\mathrm{d}t}=-5u_{1}(t)u_{2}(t)-(u_{1}(t)+7)\sin(\omega t)
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset line
LatexCommand rule
offset "0.5ex"
width "100col%"
height "1pt"

\end_inset


\end_layout

\begin_layout Standard
(Kiusalaas 7.1 Soal 11) Selesaikan persamaan diferensial dengan menggunakan
 metode numerik
\begin_inset Formula 
\[
y'=\sin(xy)
\]

\end_inset

dengan syarat awal 
\begin_inset Formula $y(x=0)=2.0$
\end_inset

, 2.5, 3.0, 3.5.
 Plot solusinya untuk 
\begin_inset Formula $0\leq x\leq10$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset CommandInset line
LatexCommand rule
offset "0.5ex"
width "100col%"
height "1pt"

\end_inset


\end_layout

\begin_layout Standard
(Kiusalaas 7.1 Soal 8) Seorang penerjun payung dengan massa 
\begin_inset Formula $m$
\end_inset

 jatuh secara vertikal dan mengalami gaya gesek 
\begin_inset Formula $F_{D}=c_{D}\dot{y}^{2}$
\end_inset

, dengan 
\begin_inset Formula $y$
\end_inset

 diukur ke arah bawah diukur dari titik awal jatuh.
 Persamaan diferensial yang menjelaskan gerakan jatuh ini adalah
\begin_inset Formula 
\[
\ddot{y}=g-\frac{c_{D}}{m}\dot{y}^{2}
\]

\end_inset

Tentukan waktu yang diperlukan untuk mencapai jarak jatuh sebesar 
\begin_inset Formula $5000$
\end_inset

 m.
 Gunakan 
\begin_inset Formula $g=9.80665$
\end_inset

 
\begin_inset Formula $\mathrm{m/s^{2}}$
\end_inset

, 
\begin_inset Formula $c_{D}=0.2028$
\end_inset

 kg/m, dan 
\begin_inset Formula $m=80$
\end_inset

 kg.
\end_layout

\end_body
\end_document